well balanced unstaggered central schemes for the euler equations with gravitation we consider the euler equations with gravitational source term and propose a new well balanced unstaggered central finite volume scheme which can preserve the hydrostatic balance state exactly the proposed scheme evolves a nonoscillatory numerical solution on a single grid avoids the time consuming process of solving riemann problems arising at the cell interfaces and is second order accurate both in time and space furthermore the numerical scheme follows a well balanced discretization that first discretizes the gravitational source term according to the discretization of the flux terms and then mimics the surface gradient method and discretizes the density and energy according to the discretization of steady state density and energy functions respectively finally several numerical experiments demonstrating the performance of the well balanced schemes in both one and two spatial dimensions are presented the results indicate that the new scheme is accurate simple and robust

fully discrete entropy conservative schemes of arbitrary order we consider weak solutions of hyperbolic or hyperbolic elliptic systems of conservation laws in one space dimension and their approximation by finite difference schemes in conservative form the systems under consideration are endowed with an entropy entropy flux pair we introduce a general approach to construct second and third order accurate fully discrete in both space and time entropy conservative schemes in general these schemes are fully nonlinear implicit but in some important cases can be explicit or linear implicit furthermore semidiscrete entropy conservative schemes of arbitrary order are presented the entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization sensitive shock waves finally specific examples are investigated and tested numerically our approach extends the results and techniques by tadmor in numerical methods for compressible flows finite difference element and volume techniques asme new york 1986 pp 149 158 lefloch and rohde siam j numer anal 37 2000 pp 2023 2060

numerical solutions to compressible flows in a nozzle with variable cross section compressible flows in a nozzle can be modeled by the gas dynamics equations in one dimensional space with source terms it turns out that along stationary waves the entropy is conserved investigating properties of the system leads us to the determination of stationary waves relying on this analysis we construct a numerical scheme which takes into account the use of stationary waves our scheme is shown to be capable of maintaining equilibrium states this demonstrates the efficiency of the new scheme over classical ones which usually give unsatisfactory results when reducing the refinement of the mesh size moreover our scheme converges much faster than the classical ones in most cases

pseudospectral algorithms for navier stokes simulation of turbulent flows in cylindrical geometry with coordinate singularities we present a new family of algorithms for incompressible 3d navier stokes equations in cylindrical geometry a model problem of turbulent flow calculation in an infinite circular pipe r z 0 r r 0 2 z is considered and used for accuracy stability and efficiency estimations algorithms are based on galerkin trigonometric approximation for uniform variables z on pseudospectral polynomial approximation in the r direction with different sets of collocation nodes and on implicit and predictor corrector time advancement schemes in all cases high infinite order spatial accuracy is retained despite the presence of coordinate singularity at r 0 to achieve this we exploit the behaviour of analytic functions of variables r z in the vicinity of r 0 we analyze the advantages and disadvantages of four navier stokes algorithms in method a a new splitting technique is developed which makes use of a second order predictor corrector scheme and nontraditional fractional step procedure stability and efficiency characteristics of this scheme exceed that of the usually used mixed adams bashforth crank nicolson time advancement to minimize errors due to splitting algorithm b is suggested that has no fractional steps in this method pressure values are eliminated from discretized navier stokes equations by means of equivalent matrix operations although conventional chebyshev collocation nodes rl r cos l 2q l 0 1 q are used in both methods the discrete boundary conditions at r 0 consistent with analytic behaviour of solutions for small r are fully accessible for the first time in addition approximations developed prevent the appearance of various pathological with e g spurious parasitic modes etc discretizations of navier stokes operators in algorithm c we propose a new set of collocation nodes rl 1 xl r 2 l 0 1 q where xl 1 1 l 1 2 q 1 are the zeros of jacobi polynomial p 2 1 0 1 x xd 1 x0 1 it is demonstrated that pseudospectral polynomial approximation with this set of nodes possesses the discrete analogue to the energy conservation law of the original navier stokes initial boundary value problem the latter is of special significance for the algorithm s nonlinear stability in method d new dependent variables are introduced that completely consider the form of the analytic pressure and velocity components at small r we show that the discrete navier stokes equations admit in this case an efficient solution procedure finally we present a technique that can be used for exhaustive a priori estimates of the algorithm s accuracy and stability characteristics in the linear approach

realizing rotation vectors for torus homeomorphisms we consider the rotation set p f for a lift f of a homeomorphism f t2 t2 which is homotopic to the identity our main result is that if a vector v lies in the interior of p f and has both coordinates rational then there is a periodic point x e t2 with the property that

a mixed finite element formulation for maxwell s equations in the time domain abstract a galerkin finite element solution technique for the maxwell s equations is discussed this new formulation can be viewed as a generalization of certain staggered grid finite difference schemes to arbitrary meshes it is shown that this technique is simple to implement and is more accurate as well as more cost effective than the standard equal order finite element approach numerical are presented to evaluate the performance of this new element relative to the standard element

arbitrarily high order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws we design arbitrarily high order accurate entropy stable schemes for systems of conservation laws the schemes termed tecno schemes are based on two main ingredients i high order accurate entropy conservative fluxes and ii suitable numerical diffusion operators involving eno reconstructed cell interface values of scaled entropy variables numerical experiments in one and two space dimensions are presented to illustrate the robust numerical performance of the tecno schemes

locally conformal finite difference time domain techniques for particle in cell plasma simulation the dey mittra s dey r mitra a locally conformal finite difference time domain fdtd algorithm for modeling three dimensional perfectly conducting objects ieee microwave guided wave lett 7 273 1997 finite difference time domain partial cell method enables the modeling of irregularly shaped conducting surfaces while retaining second order accuracy we present an algorithm to extend this method to include charged particle emission and absorption in particle in cell codes several examples are presented that illustrate the possible improvements that can be realized using the new algorithm for problems relevant to plasma simulation

rotation and entropy for a given map f x x and an observable o x llkd rotation vectors are the limits of ergodic averages of w we study which part of the topological entropy of f is associated to a given rotation vector and which part is associated with many rotation vectors according to this distinction we introduce directional and lost entropies we discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps

affordable entropy conserving and entropy stable flux functions for the ideal mhd equations in this work we design an entropy stable finite volume approximation for the ideal magnetohydrodynamics mhd equations the method is novel as we design an affordable analytical expression of the numerical interface flux function that discretely preserves the entropy of the system to guarantee the discrete conservation of entropy requires the addition of a particular source term to the ideal mhd system exact entropy conserving schemes cannot dissipate energy at shocks thus to compute accurate solutions to problems that may develop shocks we determine a dissipation term to guarantee entropy stability for the numerical scheme numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness

a stochastic galerkin method for first order quasilinear hyperbolic systems with uncertainty this paper is concerned with generalized polynomial chaos gpc approximation for first order quasilinear hyperbolic systems with uncertainty the one dimensional 1d hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the jacobian matrix then the gpc stochastic galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gpc expansion coefficients the resulting deterministic gpc galerkin system is discretized by a path conservative finite volume weno scheme in space and a third order total variation diminishing runge kutta method in time the method is further extended to two dimensional 2d quasilinear hyperbolic system with uncertainty where the symmetric hyperbolicity of the one dimensional gpc galerkin system is carried over via an operator splitting technique several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gpc stochastic galerkin method

a method for incorporating gauss lasw into electromagnetic pic codes abstract an algorithm is presented which solves a modified set of maxwell s equations none of them elliptic in an electromagnetic pic type code the algorithm prevents large violations of gauss law by diffusing away numerical errors arising from the particle to grid apportionment method there are no restrictions on the manner in which charge and current densities are determined from the particle distribution in particular point particles and simple linear interpolation are used

well balanced schemes for the euler equations with gravitation well balanced high order finite volume schemes are designed to approximate the euler equations with gravitation the schemes preserve discrete equilibria corresponding to a large class of physically stable hydrostatic steady states based on a novel local hydrostatic reconstruction the derived schemes are well balanced for any consistent numerical flux for the euler equations the form of the hydrostatic reconstruction is both very simple and computationally efficient as it requires no analytical or numerical integration moreover as required by many interesting astrophysical scenarios the schemes are applicable beyond the ideal gas law both first and second order accurate versions of the schemes and their extension to multi dimensional equilibria are presented several numerical experiments demonstrating the superior performance of the well balanced schemes as compared to standard finite volume schemes are also presented

a multiwave approximate riemann solver for ideal mhd based on relaxation ii numerical implementation with 3 and 5 waves in the first part of this work bouchut et al j comput phys 108 7 41 2007 we introduced an approximate riemann solver for one dimensional ideal mhd derived from a relaxation system we gave sufficient conditions for the solver to satisfy discrete entropy inequalities and to preserve positivity of density and internal energy in this paper we consider the practical implementation and derive explicit wave speed estimates satisfying the stability conditions of bouchut et al j comput phys 108 7 41 2007 we present a 3 wave solver that well resolves fast waves and material contacts and a 5 wave solver that accurately resolves the cases when two eigenvalues coincide a full 7 wave solver which is highly accurate on all types of waves will be described in a follow up paper we test the solvers on one dimensional shock tube data and smooth shear waves

roe matrices for ideal mhd and systematic construction of roe matrices for systems of conservation laws in this paper the construction of a roe s scheme for the conservative system of ideal magnetohydrodynamics mhd is presented as this method relies on the computation of a roe matrix the problem is to find a matrixa ul ur which satisfies the following properties it is required to be consistent with the jacobian of the fluxf to have real eigenvalues a complete set of eigenvectors and to satisfy the relation f a ul ur u whereulandurare two admissible states and utheir difference for the ideal mhd system using eulerian coordinates a roe matrix is obtained without any hypothesis on the specific heat ratio especially its construction relies on an original expression of the magnetic pressure jump moreover a roe matrix is computed for lagrangian ideal mhd by extending the results of munz who obtained such a matrix for the system of lagrangian gas dynamics so this second matrix involves arithmetic averages unlike the eulerian one which contains classical roe averages like in eulerian gas dynamics in this paper a systematic construction of lagrangian roe matrices in terms of eulerian roe matrices for a general system of conservation laws is also presented this result applied to the above eulerian and lagrangian matrices for ideal mhd gives two new matrices for this system in the same way by applying this construction to the gas dynamics equations new roe matrices are also obtained all these matrices allow the construction of roe type schemes some numerical examples on the shock tube problem show the applicability of this method

the embedded curved boundary method for orthogonal simulation meshes a method for incorporating piecewise linear approximations to curved boundaries is developed for codes using orthogonal meshes the resulting method retains much of the fidelity of adaptive triangle meshes in 2d while retaining essentially all of the speed of traditional methods relying on banded matrix solutions we find this technique useful for situations lying between the crude but fast stairstep approximation and the elegant but slower adaptive mesh techniques this method provides a practical approach to problems such as space charge emission while retaining the simplicity and run time advantage of orthogonal mesh based codes examples from space charge limited flow between curved electrodes are given

a multiwave approximate riemann solver for ideal mhd based on relaxation i theoretical framework we present a relaxation system for ideal magnetohydrodynamics mhd that is an extension of the suliciu relaxation system for the euler equations of gas dynamics from it one can derive approximate riemann solvers with three or seven waves that generalize the hllc solver for gas dynamics under some subcharacteristic conditions the solvers satisfy discrete entropy inequalities and preserve positivity of density and internal energy the subcharacteristic conditions are nonlinear constraints on the relaxation parameters relating them to the initial states and the intermediate states of the approximate riemann solver itself the 7 wave version of the solver is able to resolve exactly all material and alfven isolated contact discontinuities practical considerations and numerical results will be provided in another paper

hybrid block amr in cartesian and curvilinear coordinates mhd applications abstract we present a novel hybrid block adaptive scheme for use in solving sets of near conservation laws in general orthogonal coordinate systems the adaptive mesh refinement amr scheme is block based i e individual grids have a pre fixed number of grid cells and is implemented for any dimensionality d its hybrid character relaxes the common approach where a block that needs refinement triggers 2 d subblocks when grids are refined by a factor of 2 this introduces incomplete families in the grid hierarchy but approaches the optimal fit to developing flow features inherent in the original patch based amr strategy our hybrid block amr approach is compared with a patch based amr one both exploiting openmp parallelism the implementation is able to handle general curvilinear coordinates for which restriction and prolongation formulae are presented along with boundary treatments at singular boundaries demonstrative examples cover hydro and magnetohydrodynamic mhd applications including tests on a 2d polar grid a 2 5d spherical and a 3d cylindrical configuration the applications cover classical up to relativistic mhd simulations of particular relevance for astrophysical magnetized jet and stellar wind studies

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

a spectral time discretization for flows with dominant periodicity an accurate and efficient treatment of periodic and quasi periodic flows based on the temporal fourier decomposition of the navier stokes equations is suggested a numerical implementation for a laminar afterbody wake in a 2d channel is presented this implementation is formulated in primitive variables and uses an ordinary second order accurate finite volume space discretization combined with a standard pressure correction procedure a multistep time marching scheme for numerical and physical transients is developed for flows with a variable dominant period a period correction algorithm is used the transients characterizing the instability development are simulated the numerical results obtained for the afterbody wake confirm the expectations concerning the efficiency and high time accuracy of the method moreover the method provides direct access to quantities difficult to obtain by other methods such as the envelope and the angular velocity variation of the unstable mode

an unsplit cell centered godunov method for ideal mhd we present a second order godunov algorithm for multidimensional ideal mhd our algorithm is based on the unsplit formulation of colella j comput phys 87 1990 with all of the primary dependent variables centered at the same location to properly represent the divergence free condition of the magnetic fields we apply a discrete projection to the intermediate values of the field at cell faces and apply a filter to the primary dependent variables at the end of each time step we test the method against a suite of linear and nonlinear tests to ascertain accuracy and stability of the scheme under a variety of conditions the test suite includes rotated planar linear waves mhd shock tube problems low beta flux tubes and a magnetized rotor problem for all of these cases we observe that the algorithm is second order accurate for smooth solutions converges to the correct weak solution for problems involving shocks and exhibits no evidence of instability or loss of accuracy due to the possible presence of non solenoidal fields

the topology and dynamics of the hyperspaces of normal fuzzy sets and their inverse limit spaces given a compact and connected metric space continuum x we study topological and dynamical properties of the hyperspace of normal fuzzy sets f1 x equipped with the hausdorff endograph or sendograph metric among the many results we show that it is contractible path connected locally contractible locally path connected locally simply connected and locally connected for the endograph metric the hyperspace f1 x is a continuum and then for a topological graph x we show how using the inverse limit approach of barge and martin the inverse limit of a fuzzy dynamical system on x can be realized as an attractor of a fuzzy dynamical system on a manifold

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

an analysis of time discretization in the finite element solution of hyperbolic problems abstract the problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined a modified equation analysis reveals that the classical second order accurate time stepping algorithms i e the lax wendroff leap frog and crank nicolson methods properly combine with piecewise linear finite elements in advection problems only for small values of the time step on the contrary as the courant number increases the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit cfl property are not achieved these fundamental numerical properties can however be recovered while still remaining in the standard galerkin finite element setting by increasing the order of accuracy of the time discretization this is accomplished by exploiting the taylor series expansion in the time increment up to the third order before performing the galerkin spatial discretization using piecewise linear interpolations as a result it appears that the proper finite element equivalents of second order finite difference schemes are implicit methods of incremental type having third and fourth order global accuracy on uniform meshes taylor galerkin methods numerical results for several linear examples are presented to illustrate the properties of the taylor galerkin schemes in one and two dimensional calculations

summing logarithmic expansions for singularly perturbed eigenvalue problems strong localized perturbations of linear and nonlinear eigenvalue problems in a bounded two dimensional domain d are considered the effects on an eigenvalue lambda 0 of the lapla cian and on the fold point lambda c0 of a nonlinear eigenvalue problem of removing a small subdomain d epsilon of radius epsilon from d and imposing a condition on the boundary of the resulting hole are determined using the method of matched asymptotic expansions it is shown that the expansions of the eigenvalues and fold points for these perturbed problems start with infinite series in powers of 1 log epsilon d kappa here d kappa is a constant that depends on the shape of d epsilon and on the precise form of the boundary condition on the hole in each case it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole this related problem is not stiff and can be solved numerically

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

the origin of spurious solutions in computational electromagnetics it is commonly believed that the divergence equations in the maxwell equations are redundant for transient and timeharmonic problems therefore most of the numerical methods in computational electromagnetics solve only two first order curl equations or the second order curl curl equations this misconception is the true origin of spurious modes and inaccurate solutions in computational electromagnetics by studying the div curl system this paper clarifies that the first order maxwell equations are not overdetermined and the divergence equations must always be included to maintain the ellipticity of the system in the space domain to guarantee the uniqueness of the solution and the accuracy of the numerical methods and to eliminate the infinitely degenerate eigenvalue this paper shows that the common derivation and usage of the second order curl curl equations are incorrect and that the solution of helmholtz equations needs the divergence condition to be enforced on an associated part of the boundary the div curl method and the least squares method introduced in this paper provide rigorous derivation of the equivalent second order maxwell equations and their boundary conditions the node based least squares finite element method lsfem is recommended for solving the first order full maxwell equations directly examples of the numerical solutions by lsfem are given to demonstrate that the lsfem is free of spurious solutions

notes on the eigensystem of magnetohydrodynamics the eigenstructure of the equations governing one dimensional ideal magnetohy drodynamics is examined motivated by the wish to exploit it for construction of high resolution computational algorithms the results are given in simple forms that avoid indeterminacy or degeneracy whenever possible the unavoidable indeterminacy near the magnetosonic or triple umbilic state is analysed and shown to cause no difficulty in evaluating a numerical flux function the structure of wave paths close to this singularity is obtained and simple expressions are presented for the structure coefficients that govern wave steepening

well balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation we propose and analyze a finite volume scheme of the godunov type for conservation laws with source terms that preserve discrete steady states the scheme works in the resonant regime as well as for problems with discontinuous flux moreover an additional modification of the scheme is not required to resolve transients and solutions of non linear algebraic equations are not involved our well balanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux due to the difficulty of obtaining bv estimates we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero we include numerical experiments in order to show the features of the scheme and how it compares with a wellbalanced scheme from the literature

new rotation sets in a family of torus homeomorphisms we construct a family varphi t t in 0 1 of homeomorphisms of the two torus isotopic to the identity for which all of the rotation sets rho varphi t can be described explicitly we analyze the bifurcations and typical behavior of rotation sets in the family providing insight into the general questions of toral rotation set bifurcations and prevalence we show that there is a full measure subset of 0 1 consisting of infinitely many mutually disjoint non trivial closed intervals on each of which the rotation set mode locks to a constant polygon with rational vertices that the generic rotation set in the hausdorff topology has infinitely many extreme points accumulating on a single totally irrational extreme point at which there is a unique supporting line and that although rho varphi t varies continuously with t the set of extreme points of rho varphi t does not the family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set or by any directional ergodic measure

a continuity preserving and divergence cleaning algorithm based on purely and damped hyperbolic maxwell equations in inhomogeneous media in the numerical solution of maxwell s equations for dynamic electromagnetic fields the two gauss s laws are usually not considered since they are a natural consequence of faraday s and maxwellampre s laws if the charge conservation law is satisfied however when the charge conservation law is not satisfied the numerical errors of gauss s laws will increase unbounded leading to numerical instability or breakdown unfortunately the charge conservation law can be easily violated in self consistent waveparticle simulations in the meantime the violation of gauss s laws will also result in an increased error in the normal flux continuity in the simulations of pure electromagnetic problems the satisfaction of tangential field continuity across a material interface is sufficient to yield accurate numerical results however in a self consistent waveparticle simulation the normal components are as important as the tangential components since they are critical in predicting the particle kinetics in this paper a divergence cleaning method is presented to enforce gauss s laws and normal flux continuity by introducing auxiliary variables and damping terms into maxwell s equations in inhomogeneous media which yield hyperbolic and mixed hyperbolicparabolic maxwell equations the numerical solution schemes of the resulting purely and damped hyperbolic maxwell equations are introduced under the framework of the discontinuous galerkin time domain method it is shown through several numerical examples that the proposed method is able to preserve the continuity conditions for both tangential field components and normal flux components and is effective in cleaning the numerical errors in both gauss s laws

linearization of conservative toral homeomorphisms we give an equivalent condition for the existence of a semi conjugacy to an irrational rotation for conservative homeomorphisms of the two torus this leads to an analogue of poincare s classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property for minimal toral homeomorphisms the result extends to arbitrary dimensions further we provide a basic classification for the dynamics of toral homeomorphisms with all points non wandering

a finite volume method for the maxwell equations in the time domain a finite volume scheme for the maxwell equations is proposed using collocated nonorthogonal curvilinear grid arrangements with the advantage that all components of the electric and magnetic field are stored at the same computational location and time it is based on construction principles of high resolution schemes for hyperbolic conservation laws and takes into account the local wave propagation the implementation of boundary conditions based on the characteristic theory is described a new divergence correction technique is proposed to preserve locally the charge conservation this is important if the maxwell solver is used within an electromagnetic particle in cell pic code the accuracy and efficiency of this finite volume framework is demonstrated with problems of electromagnetic wave propagation and of the self consistent motion of charged particles

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

numerical solutions for the pressure poisson equation with neumann boundary conditions using a non staggered grid 1 abstract numerical solutions are obtained for the pressure poisson equation with neumann boundary conditions using a non staggered grid the existence of a solution for this equation requires the satisfaction of a compatibility condition which relates the source of the poisson equation and the neumann boundary conditions this compatibility condition is not automatically satisfied on non staggered grids failure to satisfy the compatibility condition leads to non convergent iterative solutions consistent finite difference approximations for the pressure equation with neumann boundary conditions are developed to satisfy the compatibility condition on non staggered grids the method is applied to calculate the pressure coefficient in a driven cavity when given the velocity field the velocity is computed from the stream function vorticity formulation of the navier stokes equations

on the divergence free condition in godunov type schemes for ideal magnetohydrodynamics the upwind constrained transport method we present a general framework to design godunov type schemes for multidimensional ideal magnetohydrodynamic mhd systems having the divergence free relation and the related properties of the magnetic field b as built in conditions our approach mostly relies on the constrained transport ct discretization technique for the magnetic field components originally developed for the linear induction equation which assures b num 0 and its preservation in time to within machine accuracy in a finite volume setting we show that the ct formalism when fully exploited can be used as a general guideline to design the reconstruction procedures of the b vector field to adapt standard upwind procedures for the momentum and energy equations avoiding the onset of numerical monopoles of o 1 size and to formulate approximate riemann solvers for the induction equation this general framework will be named here upwind constrained transport uct to demonstrate the versatility of our method we apply it to a variety of schemes which are finally validated numerically and compared a novel implementation for the mhd case of the second order roetype positive scheme by lin and lax j comput fluid dyn 5 1996 133 and both the second and third order versions of a central type mhd scheme presented by londrillo and del zanna astrophys j 530 2000 508 where the basic uct strategies have been first outlined

electromagnetics via the taylor galerkin finite element method on unstructured grids traditional techniques for computing electromagnetic solutions in the time domain rely on finite differences these so called fdtd finite difference time domain methods are usually defined only on regular lattices of points and can be too restrictive for geometrically demanding problems great geometric flexibility can be achieved by abandoning the regular latticework of sample points and adopting an unstructured grid an unstructured grid allows one to place the grid points anywhere one chooses so that curved boundaries can be fit with ease and local regions in which the field gradients are steep can be selectively resolved with a fine mesh in this paper we present a technique for solving maxwell s equations on an unstructured grid based on the taylor galerkin finite element method we present several numerical examples which reveal the fundamental accuracy and adaptability of the method although our examples are in two dimensions the techniques and results generalize readily to 3d

application of dey mittra conformal boundary algorithm to 3d electromagnetic modeling the dey mittra conformal boundary conditions have been implemented for the finite difference time domain fdtd electromagnetic solver of the vorpal plasma simulation framework and studied in the context of three dimensional large scale computations the maximum stable time step when using these boundary conditions can be arbitrarily small due to the presence of small fractional cells inside the vacuum region use of the gershgorin circle theorem allows the determination of a rigorous criterion for exclusion of small cells in order to have numerical stability for particular values of the ratio f d m dt dt c f l of the time step to the courant friedrichs lewy value for the infinite system application to a spherical cavity shows that these boundary conditions allow computation of frequencies with second order error for sufficiently small f d m however for sufficiently fine resolution dependent on f d m the error becomes first order just like the error for stair step boundary conditions nevertheless provided one does use a sufficiently small value of f d m one can obtain third order accuracy through richardson extrapolation computations for the tesla superconducting rf cavity design compare favorably with experimental measurements

exact spectrum analysis of a one dimensional hubbard model for a one dimensional hubbard model with a half filled band and one extra hole the single hole green function is calculated analytically the spectrum analysis shows that the n electrons tend to move so as to form identical and consecutive packets to satisfy rules fixed by arithmetical properties of the integer n

a positive muscl hancock scheme for ideal magnetohydrodynamics we present a highly robust second order accurate scheme for the euler equations and the ideal mhd equations the scheme is of predictor corrector type with a muscl scheme following as a special case the crucial ingredients are an entropy stable approximate riemann solver and a new spatial reconstruction that ensures positivity of mass density and pressure for multidimensional mhd a new discrete form of the powell source terms is vital to ensure the stability properties the numerical examples show that the scheme has superior stability compared to standard schemes while maintaining accuracy in particular the method can handle very low values of pressure i e low plasma b or high mach numbers and low mass densities

on digit frequencies in beta expansions we study the sets df of digit frequencies of expansions of numbers in 0 1 we show that df is a compact convex set with countably many extreme points which varies continuously with that there is a full measure collection of non trivial closed intervals on each of which df mode locks to a constant polytope with rational vertices and that the generic digit frequency set has infinitely many extreme points accumulating on a single non rational extreme point whose components are rationally independent

transfer matrix approach to the three dimensional ising model the 111 plane to 111 plane transfer matrix of the ising model on a simple cubic lattice is obtained in a simple product form by using a purpose built matrix product previously defined by this author as a first application of this theoretical result the exact analytic expression of the partition function of an ising model on a 2 2 infinity lattice is derived

functions of infinite generalized cyclic matrices analytic expressions are derived for the elements or the trace of generalized cyclic matrix functions application to the representation of the linear difference operators with constant coefficients and periodic boundary conditions is considered in order to express functions of such operators in closed form

a staggered mesh algorithm using high order godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations the equations of magnetohydrodynamics mhd have been formulated as a hyperbolic system of conservation laws in that form it becomes possible to use higher order godunov schemes for their solution this results in a robust and accurate solution strategy however the magnetic field also satisfies a constraint that requires its divergence to be zero at all times this is a property that cannot be guaranteed in the zone centered discretizations that are favored in godunov schemes without involving a divergence cleaning step in this paper we present a staggered mesh strategy which directly uses the properly upwinded fluxes that are provided by a godunov scheme the process of directly using the upwinded fluxes relies on a duality that exists between the fluxes obtained from a higher order godunov scheme and the electric fields in a plasma by exploiting this duality we have been able to construct a higher order godunov scheme that ensures that the magnetic field remains divergence free up to the computer s round off error we have even presented a variant of the basic algorithm that uses multidimensional features in the flow to design an upwinded strategy that aligns itself with the predominant upwinded direction in the flow we have devised several stringent test problems to show that the scheme works robustly and accurately in all situations in doing so we have shown that a scheme that involves a collocation of magnetic field variables that is different from the one traditionally favored in the design of higher order godunov schemes can nevertheless offer the same robust and accurate performance of higher order godunov schemes provided the properly upwinded fluxes from the godunov methodology are used in the scheme s construction

existence of non contractible periodic orbits for homeomorphisms of the open annulus in this article we consider homeomorphisms of the open annulus mathbb a mathbb r mathbb z times mathbb r which are isotopic to the identity and preserve a borel probability measure of full support focusing on the existence of non contractible periodic orbits assume f is such homeomorphism such that the connected components of the set of fixed points of f are all compact further assume that there exists check f a lift of f to the universal covering of mathbb a such that the set of fixed points of check f is non empty and that this set projects into an open topological disk of mathbb a we prove that in this setting one of the following two conditions must be satisfied 1 f has non contractible periodic points of arbitrarily large periodic or 2 for every compact set k of mathbb a there exists a constant m depending on the compact set such that if check z and check f n check z project on k then their projections on the first coordinate have distance less or equal to m

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

finite difference time domain algorithm for electromagnetic problems involving material movement the finite difference time domain fdtd method adapted for magnetic field diffusion problems is used to study the electromagnetic induction in moving materials by including motional emf in standard fdtd electromagnetic equations the material movement is implemented by continuously changing material properties in each computational cell consistent to material advection the flux corrected transport fct algorithm is used to transport magnetic field in a fixed eulerian cell a higher time step is achieved by artificially increasing permittivity of the medium this new approach is validated with standard analytical solutions for planar magnetic flux compression system and magnetic field diffusion in moving conductors with a non relativistic velocity to our knowledge this is the first approach to use fdtd method for electromagnetic problems involving material motion

ideal glm mhd about the entropy consistent nine wave magnetic field divergence diminishing ideal magnetohydrodynamics equations abstract the paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics first we show how to extend the ideal magnetohydrodynamics mhd equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics as a byproduct of these derivations we show that not all of the commonly used divergence cleaning extensions of the ideal mhd equations are thermodynamically consistent secondly we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy it includes a mechanism to control the discrete divergence error of the magnetic field by construction and is galilean invariant we implement the new high order mhd solver in the adaptive mesh refinement code flash where we compare the divergence cleaning efficiency to the constrained transport solver available in flash unsplit staggered mesh scheme

a positive conservative method for magnetohydrodynamics based on hll and roe methods the exact riemann problem solutions of the usual equations of ideal magnetohydrodynamics mhd can have negative pressures if the initial data has b 0 this creates a problem for numerical solving because in a first order finite volume conservative godunov type method one cannot avoid jumps in the normal magnetic field component even if the magnetic field was divergenceless in the three dimensional sense we show that by allowing magnetic monopoles in mhd equations and properly taking into account the magnetostatic contribution to the lorentz force an additional source term appears in faraday s law only using the harten lax vanleer hll riemann solver and discretizing the source term in a specific manner we obtain a method which is positive and conservative we show positivity by extensive numerical experimentation this mhd hll method is positive and conservative but rather diffusive thus we show how to hybridize this method with the roe method to obtain a much higher accuracy while still retaining positivity the result is a fully robust positive conservative scheme for ideal mhd whose accuracy and efficiency properties are similar to the first order roe method and which keeps b small in the same sense as powell s method as a special case a method with similar characteristics for accuracy and robustness is obtained for the euler equations as well

a block preconditioner for an exact penalty formulation for stationary mhd the magnetohydrodynamics mhd equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas this system of non self adjoint nonlinear pdes couples the navier stokes equations for fluids and maxwell s equations for electromagnetics there has been recent interest in fully coupled solvers for the mhd system because they allow for fast steady state solutions that do not require pseudo time stepping when the fully coupled system is discretized the strong coupling can make the resulting algebraic systems difficult to solve requiring effective preconditioning of iterative methods for efficiency in this work we consider a finite element discretization of an exact penalty formulation for the stationary mhd equations posed in two dimensional domains this formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a lagrange multiplier we consider extending block preconditioning techni

a lattice gas automaton capable of modeling three dimensional electromagnetic fields a lattice gas automaton lga capable of modeling maxwell s equations in three dimensions is described the automaton is a three dimensional interconnection of two dimensional lga cells with appropriate operations at the junctions between cells to include the properties of polarization a homogeneous mathematical description of the heterogeneous three dimensional automaton is provided in terms of the underlying binary variables the dynamics of the automaton conserve the scalar components of the electric and magnetic fields the implementation of the automaton on the cam 8 cellular automata machine is described the lga has been validated through calculation of resonant frequencies within various cavities the numerical results indicate the success of the automaton in analyzing three dimensional em field problems we have not proven analytically that this model reproduces maxwell s equations in the macroscopic limit as this is a topic of future study

high order well balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres wave propagation in idealized stellar atmospheres is modeled by the equations of ideal mhd together with the gravity source term the waves are modeled as small perturbations of isothermal steady states of the system we consider a formulation of ideal mhd based on the godunov powell form with an embedded potential magnetic field appearing as a parameter the equations are discretized by finite volume schemes based on approximate riemann solvers of the hll type and upwind discretizations of the godunov powell source terms local hydrostatic reconstructions and suitable discretization of the gravity source term lead to a well balanced scheme i e a scheme which exactly preserves a discrete version of the relevant steady states higher order of accuracy is obtained by employing suitable minmod eno and weno reconstructions based on the equilibrium variables to construct a well balanced scheme the resulting high order well balanced schemes are validated on a suite of numerical experiments involving complex magnetic fields the schemes are observed to be robust and resolve the complex physics well

improved vector fem solutions of maxwell s equations using grid pre conditioning the time domain vector finite element method is a promising new approach for solving maxwell s equations on unstructured triangular grids this method is sensitive to the quality or condition of the grid in this study grid pre conditioning techniques such as edge swapping laplacian smoothing and energy minimization are shown to improve the accuracy of the solution and also reduce the overall computational effort 1997 john wiley sons ltd

a novel averaging technique for discrete entropy stable dissipation operators for ideal mhd entropy stable schemes can be constructed with a specific choice of the numerical flux function first an entropy conserving flux is constructed secondly an entropy stable dissipation term is added to this flux to guarantee dissipation of the discrete entropy present works in the field of entropy stable numerical schemes are concerned with thorough derivations of entropy conservative fluxes for ideal mhd however as we show in this work if the dissipation operator is not constructed in a very specific way it cannot lead to a generally stable numerical scheme the two main findings presented in this paper are that the entropy conserving flux of ismail roe can easily break down for certain initial conditions commonly found in astrophysical simulations and that special care must be taken in the derivation of a discrete dissipation matrix for an entropy stable numerical scheme to be robust we present a convenient novel averaging procedure to evaluate the entropy jacobians of the ideal mhd and the compressible euler equations that yields a discretization with favorable robustness properties

a uniquely defined entropy stable matrix dissipation operator for high mach number ideal mhd and compressible euler simulations we describe a unique averaging procedure to design an entropy stable dissipation operator for the ideal magnetohydrodynamic mhd and compressible euler equations often in the derivation of an ent

gauss s law satisfying energy conserving semi implicit particle in cell method abstract the energy conserving semi implicit method ecsim introduced by lapenta 2017 has many advantageous properties compared to the classical semi implicit and explicit pic methods most importantly energy conservation eliminates the growth of the finite grid instability we have implemented ecsim in a different and more efficient manner than the original approach more importantly we have addressed two major shortcomings of the original ecsim algorithm there is no mechanism to enforce gauss s law and there is no mechanism to reduce the numerical oscillations of the electric field a classical approach to satisfy gauss s law is to modify the electric field and its divergence using either an elliptic or a parabolic hyperbolic correction based on the generalized lagrange multiplier method this correction however violates the energy conservation property and the oscillations related to the finite grid instability reappear in the modified ecsim scheme we invented a new alternative approach the particle positions are modified instead of the electric field in the correction step displacing the particles slightly does not change the energy conservation property while it can satisfy gauss s law by changing the charge density we found that the new gauss s law satisfying energy conserving semi implicit method gl ecsim produces superior results compared to the original ecsim algorithm in some simulations however there are still some numerical oscillations present in the electric field we attribute this to the simple finite difference discretization of the energy conserving implicit electric field solver we modified the spatial discretization of the field solver to reduce these oscillations while only slightly violating the energy conservation properties we demonstrate the improved quality of the gl ecsim method with several tests

provably positive high order schemes for ideal magnetohydrodynamics analysis on general meshes this paper proposes and analyzes arbitrarily high order discontinuous galerkin dg and finite volume methods which provably preserve the positivity of density and pressure for the ideal magnetohydrodynamics mhd on general meshes unified auxiliary theories are built for rigorously analyzing the positivity preserving pp property of numerical mhd schemes with a harten lax van leer hll type flux on polytopal meshes in any space dimension the main challenges overcome here include establishing certain relation between the pp property and a discrete divergence of magnetic field on general meshes and estimating proper wave speeds in the hll flux to ensure the pp property in the 1d case we prove that the standard dg and finite volume methods with the proposed hll flux are pp under a condition accessible by a pp limiter for the multidimensional conservative mhd system the standard dg methods with a pp limiter are not pp in general due to the effect of unavoidable divergence error in the magnetic field we construct provably pp high order dg and finite volume schemes by proper discretization of the symmetrizable mhd system with two divergence controlling techniques the locally divergence free elements and suitably discretized godunov powell source term the former technique leads to zero divergence within each cell while the latter controls the divergence error across cell interfaces our analysis reveals in theory that a coupling of these two techniques is very important for positivity preservation as they exactly contribute the discrete divergence terms which are absent in standard multidimensional dg schemes but crucial for ensuring the pp property several numerical tests further confirm the pp property and the effectiveness of the proposed pp schemes unlike the conservative mhd system the exact smooth solutions of the symmetrizable mhd system are proved to retain the positivity even if the divergence free condition is not satisfied our analysis and findings further the understanding at both discrete and continuous levels of the relation between the pp property and the divergence free constraint

slanted conducting boundaries and field emission of particles in an electromagnetic particle simulation code abstract a method to handle slanted perfectly conducting boundaries in electromagnetic particle simulation codes is described modifications to standard algorithms for the electromagnetic field advance charge and current densities particle destruction and field averaging are discussed in addition a new model for field emission from conducting boundary surfaces is described particles can be emitted from conformal or slanted surfaces and also from certain types of corners results of the new models on several problems involving slanted surfaces are presented

properties of cyclic permutation matrices on potts spin spaces and associated arithmetical polynomials exploring cyclic permutation matrices on a space of n q state spins novel arithmetical polynomials in q of degree n are introduced which satisfy interesting relations especially congruence relations zero to modulus n i e generalization to an arbitrary integer of fermat s little theorem and provide a natural representation of spectral properties with much physical significance

two coupled particle finite volume methods using delaunay voronoi meshes for the approximation of vlasov poisson and vlasov maxwell equations this paper deals with the approximation of vlasov poisson and vlasov maxwell equations we present two coupled particle finite volume methods which use the properties of delaunay voronoi meshes these methods are applied to benchmark calculations and engineering problems such as simulation of electron injector devices

an hllc type approximate riemann solver for ideal magnetohydrodynamics this paper presents a solver based on the hllc harten lax van leer contact wave approximate nonlinear riemann solver for gas dynamics for the ideal magnetohydrodynamics mhd equations written in conservation form it is shown how this solver also can be considered a modification of linde s adequate solver this approximation method is intended to resolve slow alfven and contact waves better than the original hll harten lax van leer solver compared to exact nonlinear solvers and roe s solver this new solver is computationally inexpensive in addition the method will exactly resolve isolated contacts and fast shocks the method also preserves positive density and pressure with two caveats first the numerical signal velocities the eigenvalues of the roe average matrix do not underestimate the physical signal velocities and second in a very few cases it may be required to change the wavespeeds of the riemann fan for the underlying hll method to guarantee positive pressures these conditi

handling the divergence constraints in maxwell and vlasov maxwell simulations the aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the vlasov maxwell equations and the maxwell equations with given sources these methods can be classified in two types field correction methods and source correction methods the field correction methods introduce new unknowns in the equations for which additional boundary conditions are in some cases non trivial to find the source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete gauss law that needs to be defined in accordance with the discretization of the maxwell propagation operator

a pressure based composite grid method for the navier stokes equations abstract in this work a pressure based composite grid method is developed for solving the incompressible navier stokes equations on domains composed of an arbitrary number of overlain grid blocks where a conservative internal boundary scheme is devised to ensure that global conservation is maintained issues concerning the differences between the conservative internal boundary scheme developed for the pressure correction method with a staggered grid and that commonly used for density based methods for compressible flow with nonstaggered grids are discussed an organizational scheme is developed in order to provide a general and more flexible means for handling arbitrarily overlain grid blocks applications of the composite grid method to various model problems with complex geometry are used to illustrate the characteristics of the present procedure

short note an efficient solver for the equations of resistive mhd with spatially varying resistivity we regularize the variable coefficient helmholtz equations arising from implicit time discretizations for resistive mhd in a way that leads to a symmetric positive definite system uniformly in the time step standard centered difference discretizations in space of the resulting pde leads to a method that is second order accurate and that can be used with multigrid iteration to obtain efficient solvers

forcing theory for transverse trajectories of surface homeomorphisms this paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and brouwer theory the main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations this allows us to derive new proofs for some known results as well as some new applications among which we note the following we extend franks and handel s classification of zero entropy maps of mathbb s 2 for non wandering homeomorphisms we show that if f is a hamiltonian homeomorphism of the annulus then the rotation set of f is either a singleton or it contains zero in the interior proving a conjecture posed by boyland we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms proving a first case of the franks misiurewicz conjecture we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms

edge element methods for maxwell s equations with strong convergence for gauss laws in this paper we propose and investigate some edge element approximations for three maxwell systems in three dimensions the stationary maxwell equations the time harmonic maxwell equations and the time dependent maxwell equations these approximations have three novel features first the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of nite element meshes including the stationary maxwell equations second they ensure the optimal strong convergence of the gauss laws in some appropriate norm in addition to the standard optimal convergence in energy norm under the general weak regularity assumptions that hold for both convex and non convex polyhedral domains and for the discontinuous coecients that may have large jumps across the interfaces between dierent media finally no saddle point discrete systems are needed to solve for the stationary maxwell equations unlike most existing edge element schemes

nonlinear magnetohydrodynamics simulation using high order finite elements a conforming representation composed of 2d finite elements and finite fourier series is applied to 3d nonlinear non ideal magnetohydrodynamics using a semi implicit time advance the self adjoint semi implicit operator and variational approach to spatial discretization are synergistic and enable simulation in the extremely stiff conditions found in high temperature plasmas without sacrificing the geometric flexibility needed for modeling laboratory experiments growth rates for resistive tearing modes with experimentally relevant lundquist number are computed accurately with time steps that are large with respect to the global alfven time and moderate spatial resolution when the finite elements have basis functions of polynomial degree p two or larger an error diffusion method controls the generation of magnetic divergence error convergence studies show that this approach is effective for continuous basis functions with p 2 where the number of test functions for the divergence control terms is less than the number of degrees of freedom in the expansion for vector fields anisotropic thermal conduction at realistic ratios of parallel to perpendicular conductivity is computed accurately with p 3 without mesh alignment a simulation of tearingmode evolution for a shaped toroidal tokamak equilibrium demonstrates the effectiveness of the algorithm in nonlinear conditions and its results are used to verify the accuracy of the numerical anisotropic thermal conduction in 3d magnetic topologies

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner

an entropy stable nodal discontinuous galerkin method for the resistive mhd equations part i theory and numerical verification abstract the first paper of this series presents a discretely entropy stable discontinuous galerkin dg method for the resistive magnetohydrodynamics mhd equations on three dimensional curvilinear unstructured hexahedral meshes compared to other fluid dynamics systems such as the shallow water equations or the compressible navier stokes equations the resistive mhd equations need special considerations because of the divergence free constraint on the magnetic field for instance it is well known that for the symmetrization of the ideal mhd system as well as the continuous entropy analysis a non conservative term proportional to the divergence of the magnetic field typically referred to as the powell term must be included as a consequence the mimicry of the continuous entropy analysis in the discrete sense demands a suitable dg approximation of the non conservative terms in addition to the ideal mhd terms this paper focuses on the resistive mhd equations our first contribution is a proof that the resistive terms are symmetric and positive definite when formulated in entropy space as gradients of the entropy variables which enables us to show that the entropy inequality holds for the resistive mhd equations this continuous analysis is the key for our dg discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality typically termed entropy stability our second contribution is a detailed derivation and analysis of the discretization on three dimensional curvilinear meshes the discrete analysis relies on the summation by parts property which is satisfied by the dg spectral element method dgsem with legendre gauss lobatto lgl nodes although the divergence free constraint is included in the non conservative terms the resulting method has no particular treatment of the magnetic field divergence errors which might pollute the solution quality our final contribution is the extension of the standard resistive mhd equations and our dg approximation with a divergence cleaning mechanism that is based on a generalized lagrange multiplier glm as a conclusion to the first part of this series we provide detailed numerical validations of our dgsem method that underline our theoretical derivations in addition we show a numerical example where the entropy stable dgsem demonstrates increased robustness compared to the standard dgsem