fast high order ader schemes for linear hyperbolic equations a reformulation of the ader approach arbitrary high order schemes using derivatives for linear hyperbolic pde s is presented this reformulation leads to a drastic decrease of the computational effort a formula for the construction of ader schemes that are arbitrary high order accurate in space and time is given the accuracy for some selected schemes is shown numerically for the two dimensional linearized euler equations as a mathematical model for noise propagation in the time domain in aeroacoustics

a power series method for computing singular solutions to nonlinear analytic systems given a system of analytic equations having a singular solution we show how to develop a power series representation for the solution this series is computable and when the multiplicity of the solution is small highly accurate estimates of the solution can be generated for a moderate computational cost in this paper a theorem is proven using results from several complex variables which establishes the basis for the approach then a specific numerical method is developed and data from numerical experiments are given

the time line interpolation method for large time step godunov type schemes abstract this paper describes the use of the time line interpolation procedure for the design of large time step godunov type schemes for systems of hyperbolic conservation laws in one dimension these schemes are based on a specific procedure to characterize the left and right states of the riemann problems at the cell interfaces when the courant number associated with the waves exceeds unity to do so the time line interpolation technique is used constant and linear reconstruction techniques are presented sonic or critical points are seen to be a source of difficulty in the algorithm and an appropriate treatment is proposed the algorithms are applied to the linear advection equation to the inviscid burgers equation and to the set of hyperbolic conservation laws that describe shallow water flow in one dimension these simulations show the superiority of the linear time reconstruction over the use of a constant time reconstruction when the linear reconstruction technique is used the modulus of the amplification factor of the scheme is equal to unity for all wave numbers inducing oscillations in the computed profile owing to the phase error the introduction of a slope limiter allows these oscillations to be eliminated but yields numerical diffusion thus restricting the range of applications of the scheme

the generalized riemann problem for reactive flows abstract a generalized riemann problem is introduced for the equations of reactive non viscous compressible flow in one space dimension initial data are assumed to be linearly distributed on both sides of a jump discontinuity the resolution of the singularity is studied and the first order variation in time of flow variables is given in exact form

a second order godunov type scheme for compressible fluid dynamics abstract a second order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented two related versions are proposed one lagrangian and the second direct eulerian they both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity assuming initial nonvanishing slopes on both sides while this solution is an extension of the solution to the classical riemann problem the resulting schemes are second order extensions of godunov s methods in both cases they are very simple to implement in computer codes several numerical examples are shown where the only additional mechanism is a simple monotonicity algorithm

propagator methods for plasma simulations application to breakdown accurate simulation of plasmas often requires a solution of the kinetic equation either directly by solving the boltzmann equation be or indirectly by means of particle simulations however kinetic simulations are still too computationally intensive for many large scale 3d simulations in this paper we examine the matching between a kinetic simulation and fluid models which we use in conjunction to form a hybrid plasma model of the breakdown process the kinetic model is tested for convergence with respect to mesh size dx and time step dt we then implement fluid models in an attempt to reproduce the results of the kinetic model to do this it is necessary to have a fluid model which provides accurate simulations with a wide range of dx and dt we accomplish this by means of a propagator or green s function approach the propagator method reduces to a finite difference scheme at small dx dt and gives correct results across a wide range of parameters for intermediate dx dt it is necessary to take considerable care to derive the correct propagator we apply the propagator method to two fluid models one uses parameters which are functions of the electric field and the other one uses parameters which are functions of the mean kinetic energy this version also explicitly conserves energy locally the details of the fluid models employed make a profound difference to the prediction of the breakdown

minimizing multi homogeneous bezout numbers by a local search method consider the multi homogeneous homotopy continuation method for solving a system of polynomial equations for any partition of variables the multi homogeneous bezout number bounds the number of isolated solution curves one has to follow in the method this paper presents a local search method for finding a partition of variables with minimal multi homogeneous bezout number as with any other local search method it may give a local minimum rather than the minimum over all possible homogenizations numerical examples show the efficiency of this local search method

implementation of the grp scheme for computing radially symmetric compressible fluid flows the study of radially symmetric compressible fluid flows is interesting both from the theoretical and numerical points of view spherical explosion and implosion in air water and other media are well known problems in application typical difficulties lie in the treatment of singularity in the geometrical source and the imposition of boundary conditions at the symmetric center in addition to the resolution of classical discontinuities shocks and contact discontinuities in the present paper we present the implementation of direct generalized riemann problem grp scheme to resolve this issue the scheme is obtained directly by the time integration of the fluid flows our new contribution is to show rigorously that the singularity is removable and derive the updating formulae for mass and energy at the center together with the vanishing of the momentum we obtain new numerical boundary conditions at the center which are then incorporated into the grp scheme the main ingredient is the passage from the cartesian coordinates to the radially symmetric coordinates

a staggered semi implicit discontinuous galerkin method for the two dimensional incompressible navier stokes equations in this paper we propose a new spatially high order accurate semi implicit discontinuous galerkin dg method for the solution of the two dimensional incompressible navier stokes equations on staggered unstructured curved meshes while the discrete pressure is defined on the primal grid the discrete velocity vector field is defined on an edge based dual grid the flexibility of high order dg methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non zero blocks per element for only one scalar unknown namely the pressure the method is computationally efficient since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions furthermore all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage which leads to savings in terms of computational effort for the time evolution part in this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step the method is validated for polynomial degrees up to p 3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data

a homotopy for solving general polynomial systems that respects m homogeneous structures methods for finding all solutions to a polynomial system f z 0 of n equations in n unknowns using a generic homotopy have been presented by several researchers over the last ten years in this paper we describe a new method for defining a homotophy to find all solutions to a polynomial system a major feature of this new approach is that the homotopy need not to be generic or random the practical consequences are that the homotopy may be chosen to more closely resemble the system being solved with a consequent potential for reducing the arc length of paths and exploiting special structures in the original system in particular if the system is naturally m homogeneous then the number of homotopy paths that need to be tracked is reduced since many systems that arise in applications have such a structure this represents a significant advance over previous homotopies we also provide a proof that the homotopy parameter is strictly increasing on homotopy paths as a function of arc length a new result which has implications for the choice of numerical path tracking algorithms

a direct eulerian grp scheme for relativistic hydrodynamics two dimensional case this paper develops a direct eulerian generalized riemann problem grp scheme for two dimensional 2d relativistic hydrodynamics rhd it is an extension of the grp scheme for one dimensional 1d rhds z c yang p he h z tang j comput phys 230 2011 7964 7987 and the grp scheme for the non relativistic hydrodynamics m ben artzi j q li g warnecke j comput phys 218 2006 19 43 in order to derive the direct eulerian grp scheme the local grp of the split 2d rhd equations in the eulerian formulation has to be directly resolved by using corresponding riemann invariants and rankine hugoniot jump conditions so that the crucial and delicate lagrangian treatment in the original grp scheme m ben artzi j falcovitz j comput phys 55 1984 1 32 may be avoided an important difference of resolving the grp of the split 2d rhd equations from the grp of the 1d rhd equations or the non relativistic hydrodynamical equations is coming from the fact that the flow regions across the shock or rarefaction wave in the grp of the split 2d rhd equations are nonlinearly coupled through the lorentz factor which is also built in terms of the tangential velocities it is a purely multi dimensional relativistic feature several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed 2d grp scheme

application of the grp scheme to open channel flow equations the grp generalized riemann problem scheme originally conceived for gasdynamics is reformulated for the numerical integration of the shallow water equations in channels of rectangular cross section variable width and bed profile including a friction model for the fluid channel shear stress this scheme is a second order analytic extension of the first order godunov scheme based on time derivatives of flow variables at cell interfaces resulting from piecewise linear data reconstruction in cells the second order time integration is based on solutions to generalized riemann problems at cell interfaces thus accounting for the full governing equations including source terms the source term due to variable bed elevation is treated in a well balanced way so that quiescent flow is exactly replicated this is done by adopting the surface gradient method sgm several problems of steady or unsteady open channel flow are considered including the terms corresponding to variable channel width and bed elevation as well as to shear stress at the fluid channel interface using the manning friction model in all these examples remarkable agreement is obtained between the numerical integration and the exact or accurate solutions

numerical simulation of shock wave focusing at fold caustics with application to sonic boom weak shock wave focusing at fold caustics is described by the mixed type elliptic hyperbolic nonlinear tricomi equation this paper presents a new and original numerical method for solving this equation using a potential formulation and an exact numerical solver for handling nonlinearities validation tests demonstrate quantitatively the efficiency of the algorithm which is able to handle complex waveforms as may come out from optimized aircraft designed to minimize sonic booms it provides a real alternative to the approximate method of the hodograph transform this motivated the application to evaluate the ground track focusing of sonic boom for an accelerating aircraft by coupling cfd euler simulations performed around the mock up on an adaptated mesh grid atmospheric propagation modeling and the tricomi algorithm the chosen configuration is the european eurosup mock up convergence of the focused boom at the ground level as a function of the matching distance is investigated to demonstrate the efficiency of the numerical process as a conclusion it is indicated how the present work may pave the way towards a study on sonic superboom focused boom mitigation

higher order multilevel framework for ader scheme in computational aeroacoustics abstract the versatile arbitrary derivative ader scheme is cast in a multilevel framework ml ader for fast solution of system of linear hyperbolic partial differential equations the solution is cycled through spatial operators of varying accuracy while retaining highest order accuracy by the use of a forcing function accuracy analysis of the multilevel framework including in the ml ader form is carried out in time domain as well as frequency domain results are obtained for benchmark problems in computational aeroacoustics at a much reduced computational cost

time domain solution of the parabolic equation including nonlinearity abstract this paper develops a two dimensional range and depth formulation for small angle propagation of nonlinear acoustic pulses and weak shocks in a refracting medium which can be range dependent the formulation readily extends to three dimensions the nonlinearity is most efficiently treated in the time domain so the signal is not fourier decomposed into component frequencies beginning with a second order wave equation which includes the lowest order nonlinearity an approximate first order nonlinear progressive wave equation npe is derived the npe is the nonlinear time domain counterpart of the linear frequency domain parabolic equation pe the derivation is accomplished by transforming the wave equation to a pulse following frame and perturbing about a unidirectional plane wave signal the npe manifests a natural separation of terms governing refraction diffraction spreading and nonlinear steepening numerical methods are outlined for the solution of the time domain problem calculations using the formulation developed here successfully follow the development of initially smooth pulses into n waves and the reflection of weak shocks from a caustic

solving systems of polynomial equations by bounded and real homotopy the homotopy method for solving systems of polynomial equations proposed by chow mallet paret and yorke is modified in two ways the first modification allows to keep the homotopy solution curves bounded the second one to work with real polynomials when solving a system of real equations for the first method numerical results are presented

accelerated gradient based optimization using adjoint sensitivities an electromagnetic feasible adjoint sensitivity technique em fast has been proposed recently for use with frequency domain solvers it makes the implementation of the adjoint variable approach to design sensitivity analysis straightforward while preserving the accuracy at a level comparable to that of the exact sensitivities the overhead computations associated with the estimation of the sensitivities in addition to the system analysis are due largely to the calculation of the derivatives of the system matrix here we describe the integration of the em fast with two methods for accelerated estimation of these derivatives the boundary layer concept and the broyden update we show that the broyden update approach broyden fast leads to an algorithm whose efficiency is problem independent and allows the computation of the response and its gradient through a single system analysis with practically no overhead both approaches are illustrated through the design of simple antennas using method of moments solvers

a direct eulerian grp scheme for compressible fluid flows a direct eulerian generalized riemann problem grp scheme is derived for compressible fluid flows riemann invariants are introduced as the main ingredient to resolve the generalized riemann problem grp directly for the eulerian formulation the crucial auxiliary lagrangian scheme in the original grp scheme is not necessary in the present framework the delicate sonic cases can be easily treated and the extension to multidimensional cases is obtained using the dimensional splitting technique

second order extension in space and time for a 3d cell centered lagrangian scheme abstract high energy density physics hedp flows are multi material flows characterized by strong shock waves and large changes in the domain shape due to rarefaction waves numerical schemes based on the lagrangian formalism are good candidates to model this kind of flows since the computational grid follows the fluid motion this provides accurate results around the shocks as well as a natural tracking of multi material interfaces and free surfaces the work proposed here is in continuity with the work of maire and nkonga 2009 more precisely the aim of this article is to develop robust and accurate methods for the 3d extension of the eucclhyd scheme with a second order extension based on muscl monotonic upstream centered scheme for conservation laws and grp generalized riemann problem procedures particular care is taken to preserve the symmetries and the monotonicity of the solutions the scheme robustness and accuracy are assessed on numerous lagrangian test cases for which the 3d extensions are very challenging

real gas flows in a duct a numerical scheme is presented for the solution of the euler equations of compressible flow of a real gas in a single spatial coordinate this include flow in a duct of variable cross section as well as flow with cylindrical or spherical symmetry and can prove useful when testing codes for the two dimensional equations governing compressible flow of a real gas the scheme is applied with success to a problem involving the interaction of converging and diverging cylindrical shocks for four equations of state and to a problem involving the reflection of a converging shock

solution of the generalized riemann problem for advection reaction equations we present a method for solving the generalized riemann problem for partial differential equations of the advection reaction type the generalization of the riemann problem here is twofold firstly the governing equations include nonlinear advection as well as reaction terms and secondly the initial condition consists of two arbitrary but infinitely differentiable functions an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws the solution procedure local and valid for sufficiently small times reduces the solution of the generalized riemann problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional riemann problems for homogeneous advection equations for spatial derivatives of the initial conditions we illustrate the approach via the model advection reaction equation the inhomogeneous burgers equation and the nonlinear shallow water equations with variable bed elevation

design optimization of planar structures using self adjoint sensitivity analysis a new practical approach to the s parameter sensitivity analysis of planar structures is proposed the response sensitivities are obtained with a self adjoint approach with discrete shape perturbations on non uniform method of moment grid the technique is illustrated through the s parameter sensitivity analysis of a planar printed antenna and a microwave double stub tuner the computed sensitivities are validated by comparing with the central finite difference estimates at the response level gradient based optimization is performed using the sensitivity information the performance of the optimized design is verified by simulation and by measurement

a new start system for solving deficient polynomial systems using continuation we are interested in all finite solutions of polynomial systems the start system normally used in the homotopy causes divergent continuation paths when applied to a deficient system our aim is to avoid this wasted computation by constructing a new start system the start system presented in this paper respects the m homogeneous structure of the system we want to solve

an efficient start system for multi homogeneous polynomial continuation when solving a system of polynomial equations using homotopy continuation the number of solution paths can often be significantly reduced by casting the equations in multi homogeneous form this requires a multi homogeneous start system having a full set of nonsingular solutions that can be easily computed we give a general form of such a start system that works for any multi homogeneous structure and that can be used efficiently in continuation

remark on the generalized riemann problem method for compressible fluid flows the generalized riemann problem grp method was proposed for compressible fluid flows based on the lagrangian formulation m ben artzi j falcovitz a second order godunov type scheme for compressible fluid dynamics j comput phys 55 1 1984 1 32 and a direct eulerian version was developed in m ben artzi j li g warnecke a direct eulerian grp scheme for compressible fluid flows j comput phys 28 2006 19 43 by using the concept of riemann invariants the central feature of the grp method is the resolution of centered rarefaction waves in this note we show how to use the concept of riemann invariants in order to resolve the rarefaction waves in the lagrangian coordinate system and result in the grp scheme

comparison of the generalized riemann solver and the gas kinetic scheme for inviscid compressible flow simulations the generalized riemann problem grp scheme for the euler equations and gas kinetic scheme gks for the boltzmann equation are two high resolution shock capturing schemes for fluid simulations the difference is that one is based on the characteristics of the inviscid euler equations and their wave interactions and the other is based on the particle transport and collisions the similarity between them is that both methods can use identical muscl type initial reconstructions around a cell interface and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time dependent flux function although both methods have been applied successfully to the inviscid compressible flow computations their performances have never been compared since both methods use the same initial reconstruction any difference is solely coming from different underlying mechanism in their flux evaluation therefore such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances since grp is so faithfully solving the inviscid euler equations the comparison can be also used to show the validity of solving the euler equations itself the numerical comparison shows that the grp exhibits a slightly better computational efficiency and has comparable accuracy with gks for the euler solutions in 1d case but the gks is more robust than grp for the 2d high mach number flow simulations the gks is absent from the shock instability and converges to the steady state solutions faster than the grp the grp has carbuncle phenomena likes a cloud hanging over exact riemann solvers the grp and gks use different physical processes to describe the flow motion starting from a discontinuity one is based on the assumption of equilibrium state with infinite number of particle collisions and the other starts from the non equilibrium free transport process to evolve into an equilibrium one through particle collisions the different mechanism in the flux evaluation deviates their numerical performance through this study we may conclude scientifically that it may not be valid to use the euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution to adapt the navier stokes ns equations is not valid either because the ns equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity this fact alludes to the consistency of the euler and navier stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high mach number flows the development of numerical algorithm is similar to the modeling process in deriving the governing equations but the control volume here cannot be shrunk to zero

coefficient parameter polynomial continuation we establish the theory for a very general approach to constructing homotopies for computing all geometrically isolated solutions to polynomial systems we assume that the coefficients for a polynomial system are given by parameters in engineering applications these would be derived from physical parameters since the coefficients themselves can be the parameters this is completely general we show that by considering the space of parameters instead of the space of coefficients we can exploit any special structure the coefficient parameter formulas impose on the solution set in particular we prove that we can ignore generic positive dimensional solution sets we also show that any side conditions additional equations can be used to reduce the number of paths these results unify and generalize all current approaches to polynimial continuation the effect of these new results is to greatly increase the efficiency of polynomial continuation in several engineering areas for example we establish a 36 path homotopy for a electrocardiographic application that needed 193 paths under the previous most appropriate theory also we show that a 16 path homotopy suffices for the inverse kinematics problem for 6 revolute joint manipulators a problem previously requiring 32 paths

higher order numerical model for simulation of time dependent variable density flows a numerical model first order accurate in time discretization and second order accurate in space discretization is presented for the simulation of time dependent laminar diffusion flames the numerical model uses a semi implicit scheme for time marching and incorporates flux corrected transport fct for convection terms and projection method for velocity pressure coupling a direct solver is used for solving the pressure poisson equation a time splitting method is adopted to split the transport equation into a convection equation in explicit form and a diffusion equation in implicit form the explicit convection equation allows for implementation of the fct without introducing an artificial viscosity the implicit diffusion equation removes the time step size restriction of the von neumann stability criterion theoretical assessments are derived for the order of accuracy of time difference by viewing the projection method as a lower upper decomposition and formulation is also given to establish the second order accuracy the current numerical model is used to solve the smith hutton problem and the burke schumann diffusion flame with or without external forcing of the fuel jet the results are compared with solutions using the schemes of lower order accuracy in the space discretization the comparison shows that significant numerical diffusion error exists in the solution when a first order upwind difference scheme or power law approximation is used the present study suggests that a higher order space difference scheme should be used for the simulation of time dependent laminar jet diffusion flames although a first order time difference scheme would be adequate

thermodynamical effects and high resolution methods for compressible fluid flows one of the fundamental differences of compressible fluid flows from incompressible fluid flows is the involvement of thermodynamics this difference should be manifested in the design of numerical schemes unfortunately the role of entropy expressing irreversibility is often neglected even though the entropy inequality as a conceptual derivative is verified for some first order schemes in this paper we refine the grp solver to illustrate how the thermodynamical variation is integrated into the design of high resolution methods for compressible fluid flows and demonstrate numerically the importance of thermodynamic effects in the resolution of strong waves as a by product we show that the grp solver works for generic equations of state and is independent of technical arguments

derivative riemann solvers for systems of conservation laws and ader methods in this paper we first briefly review the semi analytical method e f toro v a titarev solution of the generalized riemann problem for advection reaction equations proc roy soc london 458 2018 2002 271 281 for solving the derivative riemann problem for systems of hyperbolic conservation laws with source terms next we generalize it to hyperbolic systems for which the riemann problem solution is not available as an application example we implement the new derivative riemann solver in the high order finite volume ader advection schemes we provide numerical examples for the compressible euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes

riemann solvers for water hammer simulations by godunov method the water hammer phenomenon can be described by a 2 2 system of hyperbolic partial differential equations pdes numerical solution of these pdes using finite volume schemes is investigated herein the underlying concept of the godunov scheme is the riemann problem that must be solved to provide fluxes between the computational cells the presence of the kinetic terms in the momentum equation determines the existence of shock and rarefaction waves which influence the design of the riemann solver approximation of the expressions for the riemann invariants and jump relationships can be used to derive first and second order approximate non iterative solvers the first order approximate solver is almost 2000 times faster than the exact one but gives inaccurate predictions when the densities and celerities are low the second order approximate solver gives very accurate solutions and is 300 times faster than the exact iterative one detailed indications are provided in the appendices for the practical implementation of the riemann solvers described herein copyright 2000 john wiley sons ltd

a reduction procedure for generalized riemann problems with application to nonlinear transmission lines generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of pdes are obtained through the differential constraint method these solutions prove to be flexible enough to solve generalized riemann problems where discontinuous initial data are involved within such a theoretical framework the governing model of nonlinear transmission lines is investigated throughout

numerical solution to rarefaction or shock wave duct area change interaction conclusion it may be said that vt rj prescribed by eq 5 gives the best correlation with measurement for k and further substantiates the claim of so and hwang1 that the solution eqs 3 7 represents the best solution for self preserving turbulent round jets furthermore eqs 3 7 also represent the solutions for incompressible heated round jets if the turbulent prandtl number prt is assumed to be constant then eq 3 with the exponent r 2 i2 replaced by ri2pr vn2 becomes the solution of the temperature equation since eqs 1 and 2 are equally applicable for incompressible heated round jets eqs 8 11 also represent solutions for such flows therefore the present results are just as valid for nonisothermal round jets

finite volume evolution galerkin methods for hyperbolic systems the subject of the paper is the derivation and analysis of new multidimensional high resolution finite volume evolution galerkin fveg schemes for systems of nonlinear hyperbolic conservation laws our approach couples a finite volume formulation with approximate evolution operators the latter are constructed using the bicharacteristics of the multidimensional hyperbolic system such that all of the infinitely many directions of wave propagation are taken into account in particular we propose a new fveg scheme which is designed in such a way that for a linear wave equation system the approximate evolution operator calculates any one dimensional planar wave exactly this operator improves the stability of the fveg scheme considerably leading to a stability limit closer to 1 using the results obtained for the wave equation system a new approximate evolution operator for the linearized euler equations is also derived the integrals over the cell interfaces also need to be approximated with care in this case our choice of simpson s rule is guided by stability analysis of model problems second order resolution is obtained by means of a piecewise bilinear recovery numerical experiments confirm the accuracy and multidimensional behavior of the new scheme

high order upwind flux correction methods for hyperbolic conservation laws abstract totally one sided first order and second order schemes are presented employing a numerically calculated characteristic speed direction and are combined into a transportive monotonicity preserving hybrid scheme using the method of flux correction the first order scheme is free of expansion shocks and artificial extrema the hybrid scheme computes a provisional update from the first order scheme and then filters the second order corrections to prevent occurrence of new extrema computed versus analytic results are compared for two different n wave shocks and for a third case involving linear advection of a square wave results are given with and without the second order correction the second order results are always superior to first order results with the most dramatic difference occurring in the case of linear advection the results suggest that higher than second order upwind differences could be substituted in the hybrid scheme to reduce truncation error even further

a third order accurate direct eulerian grp scheme for the euler equations in gas dynamics the paper proposes and implements a third order accurate direct eulerian generalized riemann problem grp scheme for one and two dimensional 1d 2d euler equations in gas dynamics it is an extension of the second order accurate grp scheme proposed in ben artzi et al 2006 5 the approximate states in numerical fluxes of the third order accurate grp scheme are derived by using the higher order weno reconstruction of the initial data the limiting values of the time derivatives of the solutions at the singularity point and the jacobian matrix besides the limiting values of the first order time derivatives of fluid variables the second order time derivatives are also needed in developing the present grp scheme and obtained by directly and analytically resolving the local grp in the eulerian formulation via two main ingredients i e the riemann invariants and rankine hugoniot jump conditions unfortunately for the sonic case that the transonic rarefaction wave appears in the grp the jacobian matrix is singular on the sonic line to this end those approximate states are given in a different way that is based on the analytical resolution of the transonic rarefaction wave and the local quadratic polynomial interpolation the 2d grp scheme is implemented by using the third order accurate time splitting method several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed grp scheme in comparison to the second order accurate grp scheme

computation of reactive duct flows in external fields abstract a grp scheme is introduced for the numerical integration of the euler system of equations of compressible reactive flow in a duct of variable cross section subject to an external potential the grp generalized riemann problem scheme is based on an analytic solution of the grp at jump discontinuities it is a second order scheme generalizing the first order godunov scheme having the property of high resolution of shocks and other discontinuities some numerical examples are considered including an infinite spherical reflected shock a spherical blast wave and gas collapse under an external potential

towards a more accurate flux corrected transport algorithm abstract an algorithm is discussed for solving first order hyperbolic equations whose solution may exhibit very steep and changing gradients shock like and large dynamic range in developing the algorithm emphasis has been placed on obtaining more accurate solutions the algorithm is based on flux corrected transport techniques

numerical simulation of sonic boom focusing a numerical method is presented to simulate the focusing of sonic booms in the vicinity of caustics the pressure satisfies the nonlinear tricomi equation to solve this equation an iterative algorithm based on an unsteady version of the equation is used the algorithm is a modification of the pseudospectral code used for solving the zabolotskaya khokhlov kz equation in the linear case the code is validated by comparison with analytical solutions for an incoming n wave the shape of the outgoing wave observed in flight tests is recovered in the nonlinear case no analytical solution is known to compare with the numerical output validation of the numerical scheme is completed by means of four different tests first comparisons with the linear case shows that the numerical solutions behave as expected from the physics second the solution after convergence is proved to be independent of the initial guess third the maximal signal amplitude is proved to converge while increasing the discretization finally the numerical scheme is checked against the nonlinear guiraud s scaling law in the last part an application to the focusing of concorde sonic boom in acceleration is presented and ways of reducing focused booms are discussed

a homotopy for solving polynomial systems this paper presents a continuation method for finding all solutions to polynomial systems it features a simpler homotopy than has been previously published

brain venous haemodynamics neurological diseases and mathematical modelling a review behind medicine m is physiology p behind physiology is physics p and behind physics is always mathematics m for which i expect that the symmetry of the quadruplet mppm will be compatible with the characteristic bias of hyperbolic partial differential equations a theme of this paper i start with a description of several idiopathic brain pathologies that appear to have a strong vascular dimension of which the most prominent example considered here is multiple sclerosis the most common neurodegenerative disabling disease in young adults other pathologies surveyed here include retinal abnormalities transient global amnesia transient monocular blindness meniere s disease and idiopathic parkinson s disease it is the hypothesised vascular aspect of these conditions that links medicine to mathematics through fluid mechanics in very complex networks of moving boundary blood vessels the second part of this paper is about mathematical modelling of the human cardiovascular system with particular reference to the venous system and the brain a review of a recently proposed multi scale mathematical model then follows consisting of a one dimensional hyperbolic description of blood flow in major arteries and veins coupled to a lumped parameter description of the remaining main components of the human circulation derivation and analysis of the hyperbolic equations is carried out for blood vessels admitting variable material properties and with emphasis on the venous system a much neglected aspect of cardiovascular mathematics veins unlike their arterial counterparts are highly deformable even collapsible under mild physiological conditions we address mathematical and numerical challenges regarding the numerical analysis of the hyperbolic pdes we deploy a modern non linear finite volume method of arbitrarily high order of accuracy in both space and time the ader methodology in vivo validation examples and brain haemodynamics computations are shown we also point out two preliminary but important new findings through the use of mathematical models namely that extracranial venous strictures produce chronic intracranial venous hypertension and that augmented pressure increases the blood vessel wall permeability

arbitrary high order discontinuous galerkin schemes based on the grp method for compressible euler equations combined with the discontinuous galerkin dg framework the generalized riemann problem grp method is applied to design a grp dg scheme with high order accuracy for compressible euler equations since numerical fluxes with second order accuracy in time are derived by the grp method the reconstruction steps for physical variables in the new scheme are halved compared with the traditional runge kutta discontinuous galerkin rk dg scheme the numerical results are also improved due to more introduced physical information several numerical examples verify the validity of the proposed schemes

accelerated gradient based optimization of planar circuits we demonstrate for the first time the integration of adjoint variable method avm sensitivities and the method of moments mom within an optimization framework where the s parameters are used to define the cost function the avm efficiently estimates the sensitivities of the s parameters using only one mom simulation with some additional overhead two possible approaches are used the adjoint finite difference approach and the adjoint broyden update approach the avm s parameter sensitivities are successfully utilized to optimize a band stop filter a band pass filter and an electromagnetically coupled microstrip yagi antenna the results show that the optimizers supported by avm sensitivities are capable of reaching feasible designs satisfying the specifications

flux corrected pseudospectral method for scalar hyperbolic conservation laws abstract the pseudospectral method has under used advantages in problems involving shocks and discontinuities these emerge from superior accuracy in phase and group velocities as compared to finite difference schemes of all orders dispersion curves for finite difference schemes suggest that group velocity error typically outranks gibbs error as a cause of numerical oscillation a flux conservative form of the pseudospectral method is derived for compatibility with flux limiters used to preserve monotonicity the resulting scheme gives high quality results in linear advection and shock formation propagation examples

analytical shape derivatives of the mfie system matrix discretized with rwg functions an analytical formula for the shape derivative of the magnetic field integral equation mfie method of moments mom system matrix or impedance matrix is derived and validated against finite difference formulas the motivation for computing the shape derivatives stems from adjoint variable methods avm the galerkin system matrix is constructed by means of rao wilton glisson rwg basis and testing functions the shape derivative formula yields an integral representation which is of same singularity order as the integrals appearing in the traditional mfie system matrix

an entropy satisfying muscl scheme for systems of conservation laws for the high order numerical approximation of hyperbolic systems of conservation laws we propose to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by harten for scalar equations based on this new criterion we derive entropy diminishing projections that ensure both the second order of accuracy and all of the classical discrete entropy inequalities the resulting scheme is a nonlinear version of the classical van leer s muscl scheme strong convergence of this second order entropy satisfying scheme is proved for systems of two equations numerical tests demonstrate the interest of our theory