high resolution schemes for conservation laws with locally varying time steps we develop upwind methods which use limited high resolution corrections in the spatial discretization and local time stepping for forward euler and second order time discretizations l infty stability is proven for both time stepping schemes for problems in one space dimension these methods are restricted by a local cfl condition rather than the traditional global cfl condition allowing local time refinement to be coupled with local spatial refinement numerical evidence demonstrates the stability and accuracy of the methods for problems in both one and two space dimensions

local time stepping in runge kutta discontinuous galerkin finite element methods applied to the shallow water equations abstract large geophysical flows often encompass both coarse and highly resolved regions approximating these flows using shock capturing methods with explicit time stepping gives rise to a courant friedrichs lewy cfl time step constraint even if the refined regions are sparse they can restrict the global cfl condition to very small time steps vastly increasing computational effort over the whole domain one method to cope with this problem is to use locally varying time steps over the domain these are also referred to as multi rate methods in the ode literature ideally such methods must be conservative accurate and easy to implement in this study we derive a second order local time stepping procedure within a runge kutta discontinuous galerkin rkdg framework to solve the shallow water equations this procedure is based on previous first order work of the second author and collaborator kirby 1 2 3 as we are interested in both coastal and overland flows due to e g rainfall wetting and drying is incorporated into the model numerical results are shown which verify the accuracy and efficiency of the approach compared to using a globally defined cfl time step and the application of the method to rainfall runoff scenarios

the arithmetic cosine transform exact and approximate algorithms in this paper we introduce a new class of transform method the arithmetic cosine transform act we provide the central mathematical properties of the act necessary in designing efficient and accurate implementations of the new transform method the key mathematical tools used in the paper come from analytic number theory in particular the properties of the riemann zeta function additionally we demonstrate that an exact signal interpolation is achievable for any block length approximate calculations were also considered the numerical examples provided show the potential of the act for various digital signal processing applications

a collection of problems for which gaussian elimination with partial pivoting is unstable a significant collection of two point boundary value problems is shown to give rise to linear systems of algebraic equations on which gaussian elimination with row partial pivoting is unstable when standard solution techniques are used

a comparison of implicit time integration methods for nonlinear relaxation and diffusion several time integration methods for nonlinear systems are compared all of the time discretizations are based on the method but differ in their treatment of the implicit nonlinear terms one method converges the implicit nonlinear terms to a small tolerance and is often referred to as nonlinearly consistent nc newton s method or its approximation newton krylov is used to converge the nonlinearities the other methods considered are linearized and comparisons are made for a relaxation problem and a radiation diffusion problem the linearized one step method that uses the full jacobian is shown to have similar accuracy as nc methods the lagged linearization method and an extension that is second order accurate are also studied a trunction error analysis complements the numerical results for the relaxation problem it is shown that each of the second order accurate linearized methods may be more accurate than an nc method depending on the degree of nonlinearity in the problem for the radiation diffusion problem in general the nc method is most accurate and allows a larger time step however the linearized methods perfom surprisingly well

iterative linear solvers in a 2d radiation hydrodynamics code computer codes containing both hydrodynamics and radiation play a central role in simulating both astrophysical and inertial confinement fusion icf phenomena a crucial aspect of these codes is that they require an implicit solution of the radiation diffusion equations we present in this paper the results of a comparison of five different linear solvers on a range of complex radiation and radiation hydrodynamics problems the linear solvers used are diagonally scaled conjugate gradient gmres with incomplete lu preconditioning conjugate gradient with incomplete cholesky preconditioning multigrid and multigrid preconditioned conjugate gradient these problems involve shock propagation opacities varying over 5 6 orders of magnitude tabular equations of state and dynamic ale arbitrary lagrangian eulerian meshes we perform a problem size scalability study by comparing linear solver performance over a wide range of problem sizes from 1000 to 100 000 zones the fundamental question we address in this paper is is it more efficient to invert the matrix in many inexpensive steps like diagonally scaled conjugate gradient or in fewer expensive steps like multigrid in addition what is the answer to this question as a function of problem size and is the answer problem dependent we find that the diagonally scaled conjugate gradient method performs poorly with the growth of problem size increasing in both iteration count and overall cpu time with the size of the problem and also increasing for larger time steps for all problems considered the multigrid algorithms scale almost perfectly i e the iteration count is approximately independent of problem size and problem time step for pure radiation flow problems i e no hydrodynamics we see speedups in cpu time of factors of 15 30 for the largest problems when comparing the multigrid solvers relative to diagonal scaled conjugate gradient for the incomplete factorization preconditioners we see a weak dependence of iteration count on problem size the speedups observed for pure radiation flow are typically on the order of 10 relative to diagonal scaled conjugate gradient for radiation hydrodynamics problems we again see multigrid scaling perfectly however for the problems considered we see speedups relative to diagonal scaled conjugate gradient of no more than 10 with incomplete cholesky in fact either equaling or outperforming multigrid we trace these observations to the time step control and the feature of ale to relax distorted zones

searching for rare growth factors using multicanonical monte carlo methods the growth factor of a matrix quantifies the amount of potential error growth possible when a linear system is solved using gaussian elimination with row pivoting while it is an easy matter n j higham and d j higham siam j matrix anal appl 10 1989 pp 155 164 to construct examples of n times n matrices having any growth factor up to the maximum of 2 n 1 the weight of experience and analysis n j higham accuracy and stability of numerical algorithms siam philadelphia 1996 l n trefethen and r s schreiber siam j matrix anal appl 11 1990 pp 335 360 l n trefethen and i d bau numerical linear algebra siam philadelphia 1997 suggest that matrices with exponentially large growth factors are exceedingly rare here we show how to conduct numerical experiments on random matrices using a multicanonical monte carlo method to explore the tails of growth factor probability distributions our results suggest for example that the occurrence of an 8 times 8 matrix with a growth factor of 40 is on the order of a once in the age of the universe event

a vlsi architecture for simplified arithmetic fourier transform algorithm the arithmetic fourier transform aft is a number theoretic approach to fourier analysis which has been shown to perform competitively with the classical fft in terms of accuracy complexity and speed theorems developed by i s reed et al 1990 for the aft algorithm are used here to derive the original aft algorithm which bruns found in 1903 this is shown to yield an algorithm of less complexity and of improved performance over certain aft algorithms a vlsi architecture is suggested for this simplified aft algorithm this architecture uses a butterfly structure which reduces the number of additions by 25 of that used in the direct method

asynchronous discrete event schemes for pdes abstract a new class of asynchronous discrete event simulation schemes for advection diffusion reaction equations is introduced based on the principle of allowing quanta of mass to pass through faces of a regular structured cartesian finite volume grid the timescales of these events are linked to the flux on the face the resulting schemes are self adaptive and local in both time and space experiments are performed on realistic physical systems related to porous media flow applications including a large 3d advection diffusion equation and advection diffusion reaction systems the results are compared to highly accurate reference solutions where the temporal evolution is computed with exponential integrator schemes using the same finite volume discretisation this allows a reliable estimation of the solution error our results indicate a first order convergence of the error as a control parameter is decreased and we outline a framework for analysis

an implicit explicit runge kutta chebyshev scheme for diffusion reaction equations an implicit explicit imex extension of the explicit runge kutta chebyshev rkc scheme designed for parabolic pdes is proposed for diffusion reaction problems with severely stiff reaction terms the imex scheme treats these reaction terms implicitly and diffusion terms explicitly within the setting of linear stability theory the new imex scheme is unconditionally stable for reaction terms having a jacobian matrix with a real spectrum for diffusion terms the stability characteristics remain unchanged a numerical comparison for a stiff nonlinear radiation diffusion problem between an rkc solver an imex rkc solver and the popular implicit bdf solver vodpk using the krylov solver gmres illustrates the excellent performance of the new scheme

a posteriori analysis of a multirate numerical method for ordinary differential equations in this paper we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model the main purpose of this paper is to present a hybrid a priori a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations the use of only a finite number of iterations in the iterative solution of the discrete equations the numerical error arising in the solution of each component and the effects on stability arising from the multirate solution the hybrid estimate has the form of a computable a posteriori leading order expression and a provably higher order a priori expression we support this estimate by an a priori convergence analysis we present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate

algorithms for almost block diagonal linear systems codes for solving systems of ordinary differential equations for use in the method of lines for partial differential equations pdes usually provide only a banded system solver in this context a frequently occurring structure is almost block diagonal abd solving abd systems by imposing banded structure introduces fill in and is inefficient though robust efficient abd software has been developed and used in packages for solving boundary value problems with separated boundary conditions for ordinary differential equations bvodes it has not been generally exploited in pde software the situation with bordered almost block diagonal system software for bvodes with nonseparated boundary conditions is less satisfactory this survey draws on material from a variety of sources particularly p amodio et al numer linear algebra appl 7 2000 pp 275 317 and b garrett and i gladwell j comput methods sci engrg 1 2001 pp 75 98 and the references therein

a new non linear two time level central leapfrog scheme in staggered conservation flux variables for fluctuating hydrodynamics equations with gpu implementation abstract for solving landau lifshitz navier stokes fluctuating hydrodynamics equations a two time level modification of the classical central leapfrog scheme with non linear flux correction is developed the new algorithm is simple for implementation and demonstrates high accuracy in satisfying the fluctuation dissipation theorem the numerical results for equilibrium and non equilibrium problems in one dimensional and three dimensional settings are compared with theoretical predictions the results from direct molecular dynamics simulations and also with those of several popular computational schemes in the literature because of simplicity and locality of the computational stencil the new algorithm can be efficiently implemented on the gpu which accelerates the calculation in comparison with a single cpu core computation by a factor of 300

solving the advection diffusion equation on unstructured meshes with discontinuous mixed finite elements and a local time stepping procedure explicit schemes are known to provide less numerical diffusion in solving the advection diffusion equation especially for advection dominated problems traditional explicit schemes use fixed time steps restricted by the global cfl condition in order to guarantee stability this is known to slow down the computation especially for heterogeneous domains and or unstructured meshes to avoid this problem local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local cfl condition have been developed in this paper a local time stepping approach is used with a numerical model based on discontinuous galerkin mixed finite element methods to solve the advection diffusion equation the developments are detailed for general unstructured triangular meshes numerical experiments are performed to show the efficiency of the numerical model for the simulation of i the transport of a solute on highly unstructured meshes and ii density driven flow where the velocity field changes at each time step the model gives stable results with significant reduction of the computational cost especially for the non linear problem moreover numerical diffusion is also reduced for highly advective problems copyright 2009 john wiley sons ltd

an efficient local time stepping scheme for solution of nonlinear conservation laws we develop an efficient local time stepping algorithm for the method of lines approach to numerical solution of transient partial differential equations the need for local time stepping arises when adaptive mesh refinement results in a mesh containing cells of greatly different sizes the global cfl number and hence the global time step are defined by the smallest cell size this can be inefficient as a few small cells may impose a restrictive time step on the whole mesh a local time stepping scheme allows us to use the local cfl number which reduces the total number of function evaluations the algorithm is based on a second order runge kutta time integration its important features are a small stencil and the second order accuracy in the l 2 and l norms

a time accurate explicit multi scale technique for gas dynamics abstract we present a new time accurate algorithm for the explicit numerical integration of the compressible euler equations of gas dynamics this technique is based on the discrete event simulation des methodology for nonlinear flux conservative pdes y a omelchenko h karimabadi self adaptive time integration of flux conservative equations with sources j comput phys 216 1 2006 179 194 des enables adaptive distribution of cpu resources in accordance with local time scales of the underlying numerical solution it distinctly stands apart from multiple local time stepping algorithms in that it requires neither selecting a global synchronization time step nor pre determining a sequence of time integration operations for individual parts of a heterogeneous numerical system in this paper we extend the des methodology in three important directions i we apply des to a system of coupled gas dynamics equations discretized via a central upwind scheme a kurganov e tadmor new high resolution central schemes for nonlinear conservation laws and convection diffusion equations j comput phys 160 2000 241 282 a kurganov s noelle g petrova semidiscrete central upwind schemes for hyperbolic conservation laws and hamilton jacobi equations siam j sci comput 23 3 2001 707 740 ii we introduce a new preemptive event processing pep technique which automatically enforces synchronous execution of events with sufficiently close update times iii we significantly improve the accuracy of the previous algorithm y a omelchenko h karimabadi self adaptive time integration of flux conservative equations with sources j comput phys 216 1 2006 179 194 by applying locally second order in time flux conserving corrections to the solution obtained with the forward euler scheme the performance of the new technique is demonstrated in a series of one dimensional gas dynamics test problems by comparing numerical solutions obtained in event driven and equivalent time stepping simulations

linear system solvers for boundary value odes abstract we investigate the stability properties of several linear system solvers for solving boundary value odes we consider the compactification algorithm gaussian elimination with row partial pivoting and a qr algorithm applied to linear systems arising from solving bvps for which the matrix is block bidiagonal except for bordering along the last n rows and columns we will particularly compare auto s original linear solver an lu decomposition with partial pivoting and our implementation of the analogous qr algorithm to auto two other factors the underlying continuation strategy and mesh selection strategy may affect the stability of the linear system solver for ode continuation codes as well and are also discussed in our numerical investigations

time step size selection for radiation diffusion calculations the purpose of this note is to describe a time step control technique as applied to radiation diffusion standard practice only provides a heuristic criteria related to the relative change in the dependent variables the authors propose an alternative based on relatively simple physical principles this time step control applies to methods of solution that are unconditionally stable and converges nonlinearities within a time step in the governing equations commonly nonlinearities in the governing equations are evaluated using existing old time data the authors refer to this as the semi implicit si method when a method converges nonlinearities within a time step the entire governing equation including all nonlinearities is self consistently evaluated using advance time data with appropriate time centering for accuracy

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

a note on the computational complexity of the arithmetic fourier transform it is shown that the number of data points the arithmetic fourier transform aft needs for an n point fourier transform is proportional to n sup 2 thus for example while a standard fast fourier transform algorithm requires 1024 samples to yield 1024 spectral components aft would take more than 300000 samples to do the same job

event driven hybrid particle in cell simulation particle in cell models have become standard computational tools for studying complex nonlinear phenomena in space and laboratory plasmas these simulations are normally very compute intensive since they require time integration of strongly coupled equations governing the field and particle dynamics as a result despite a significant progress in hardware technology particle in cell codes are rarely used to simulate long time evolution of large scale systems with strongly varying temporal and spatial scales we propose an alternative paradigm to time stepping which is traditionally used for time integration of such systems this new approach is based on explicit discrete event simulation technology it offers distinct advantages over synchronous time stepping i updates of individual macro particles and discrete field elements are performed asynchronously ii local time increments are determined and self adaptively adjusted in time through scheduling and execution of physically meaningful local updates events the event driven time advance is accurate free of the global courant condition stable parallelizable extendable to multiple dimensions and well suited for nonuniform spatial meshes we demonstrate the new method on a one dimensional hybrid particle in cell model with applications to several plasma discontinuities including a high mach number fast magnetosonic shock and the associated plasma turbulence

the growth factor and efficiency of gaussian elimination with rook pivoting abstract gaussian elimination is among the most widely used tools in scientific computing gaussian elimination with partial pivoting requires only o n2 comparisons beyond the work required in gaussian elimination with no pivoting but can in principle have error growth that is exponential in the matrix size n gaussian elimination with complete pivoting on the other hand cannot have exponential error growth but requires o n3 comparisons beyond the work required by gaussian elimination with no pivoting numerical experiments suggest that gaussian elimination with rook pivoting is between partial pivoting and complete pivoting in terms of efficiency and accuracy in this paper we prove that rook pivoting cannot have exponential error growth we also introduce a combination of partial pivoting and rock pivoting which we call gaussian elimination with partial rook pivoting and we prove the partial rook pivoting cannot have exponential error growth we include numerical experiments showing that on a serial computer the run times for cook pivoting are almost always close to those of partial pivoting and the run times for partial rook pivoting appear to be the same as those of partial pivoting

multi adaptive galerkin methods for odes i we present multi adaptive versions of the standard continuous and discontinuous galerkin methods for odes taking adaptivity one step further we allow for individual time steps order and quadrature so that in particular each individual component has its own time step sequence this paper contains a description of the methods an analysis of their basic properties and a posteriori error analysis in the accompanying paper a logg siam j sci comput submitted we present adaptive algorithms for time stepping and global error control based on the results of the current paper

implicit explicit parallel asynchronous solver of parabolic pdes this paper presents a method for the solution of parabolic pdes on parallel computers which is a combination of implicit and explicit finite difference schemes based on a domain decomposition dd strategy moreover this method is asynchronous i e no explicit synchronization is required among processors we determine the values at subdomains boundaries by our new high order asynchronous explicit schemes then any known high order implicit finite difference scheme can be applied within each subdomain we present a technique for derivation of appropriate asynchronous explicit schemes based on green s functions synchronous versions of these schemes are obtained as special cases the applicability of this method is also demonstrated for a family of nonlinear problems our new explicit schemes are of high order and yet stable for a large time step as established in our analysis of their numerical properties moreover these schemes provide attractive properties for parallel implementation being asynchronous they allow local time stepping thus eliminating the need for a global synchronized time step moreover our asynchronous computation is time stabilizing in the sense that the calculation implicitly prevents a growing time gap between neighboring subdomains the locality property due to the exponential decay of green s functions implies that communication is needed only between neighboring processors hence this method which is designed to minimize the overhead associated with the synchronization of the multiple processors is specifically suitable for parallel computers having a high synchronization cost or highly varying load even in cases in which some processors have persistent speed differences furthermore the implementation of different resolution in each subdomain e g irregular or unstructured grid makes it valuable as an adaptive algorithm the above schemes were implemented and tested on the shared memory multi user cray j90 and sequent balance machines these implementations prove high accuracy and high degree of parallelism this work is complementary to our previous work on asynchronous schemes comput math appl 24 1992 pp 33 53 appl numer math 12 1993 pp 27 45 numer algorithms 6 1994 pp 275 296 numer algorithms 12 1996 pp 159 192

growth factor and expected growth factor of some pivoting strategies several definitions of growth factors for gaussian elimination are compared some new pivoting strategies intermediate between partial pivoting and rook pivoting are introduced for random matrices an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed a stationary behavior of the expected growth factors of the new pivoting strategies is observed bounds for the growth factors of these pivoting strategies are provided it is also shown that partial pivoting by columns produces small growth factors for matrices appearing in practical observations and for which the growth factors produced by partial pivoting are very large

efficient algorithm for 2 d arithmetic fourier transform this article presents an efficient algorithm for the two dimensional 2 d arithmetic fourier transform aft based on the mobius inversion formula of odd number series it requires fewer multiplications and has less complexity over previous algorithms in addition a technique is proposed to carry out the on axis fourier coefficients a parallel vlsi architecture is developed for the new algorithm

an adaptive multiresolution scheme with local time stepping for evolutionary pdes we present a fully adaptive numerical scheme for evolutionary pdes in cartesian geometry based on a second order finite volume discretization a multiresolution strategy allows local grid refinement while controlling the approximation error in space for time discretization we use an explicit runge kutta scheme of second order with a scale dependent time step on the finest scale the size of the time step is imposed by the stability condition of the explicit scheme on larger scales the time step can be increased without violating the stability requirement of the explicit scheme the implementation uses a dynamic tree data structure numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid fully adaptive three dimensional computations for reaction diffusion equations illustrate the additional speed up of the local time stepping for a thermo diffusive flame instability

convergence and instability in pcg methods for bordered systems abstract bordered almost block diagonal systems arise from discretizing a linearized first order system of n ordinary differential equations in a two point boundary value problem with nonseparated boundary conditions the discretization may use spline collocation finite differences or multiple shooting after internal condensation if necessary the bordered almost block diagonal system reduces to a standard finite difference structure which can be solved using a preconditioned conjugate gradient method based on a simple matrix splitting technique this preconditioned conjugate gradient method is guaranteed to converge in at most 2n 1 iterations we exhibit a significant collection of two point boundary value problems for which this preconditioned conjugate gradient method is unstable and hence convergence is not achieved

self adaptive time integration of flux conservative equations with sources we present a novel flux conserving asynchronous method for the explicit time integration of multi scale flux conservative partial differential equations with source terms unlike the conventional explicit and implicit integration schemes it is based on a discrete event simulation paradigm which describes time advance in terms of increments to physical quantities and causality rules rather than time stepping this method exerts self adaptive control over local update rates of solution by predicting and correcting changes to simulation variables in accordance with local physical scales the discrete event simulation paradigm is independent of the underlying spatial mesh and thus can be incorporated into block structured and unstructured mesh refinement techniques the effectiveness and robustness of the new method is demonstrated on a number of one dimensional uniform mesh models based on diffusion convection reaction equations the event driven integration reduces numerical approximation errors due to large local time derivatives prevents explosive numerical instabilities in locally super courant calculations and automatically reduces the cpu overhead associated with stiff terms and inactive regions in computation space

an exponential integrator for advection dominated reactive transport in heterogeneous porous media we present an exponential time integrator in conjunction with a finite volume discretisation in space for simulating transport by advection and diffusion including chemical reactions in highly heterogeneous porous media representative of geological reservoirs these numerical integrators are based on the variation of constants solution and solving the linear system exactly this is at the expense of computing the exponential of the stiff matrix comprising the finite volume discretisation using real leja points or a krylov subspace technique compared to standard finite difference based time integrators we observe for a variety of example applications that numerical solutions with exponential methods are generally more accurate and require less computational cost they hence comprise an efficient and accurate method for simulating non linear advection dominated transport in geological formations

stable parallel elimination for boundary value odes a parallelizable and vectorizable algorithm for solving linear algebraic systems arising from two point boundary value odes is described the method is equivalent to gaussian elimination with row partial pivoting applied to a certain column reordered version of the usual almost block diagonal coefficient matrix we present analytical and numerical evidence to show that the algorithm is stable in most circumstances results from implementation on a shared memory multiprocessor and a vector processor are given the approach can be extended to handle problems with multipoint and integral conditions or with algebraic parameters

inverse z transform by mobius inversion and the error bounds of aliasing in sampling a general algorithm based on two special mobius inversion formulae is developed to compute the inverse z transform this approach to fourier analysis uses what is called the arithmetic fourier transform aft with the new aft algorithm one can compute the inverse z transform of an infinite two sided sequence it is compared with the conventional dft approach both methods have aliasing errors due to sampling the error bounds of the aliasing effects in the dft and the new proposed method are established and compared in general the aft algorithm is not so vulnerable to the aliasing errors in the high frequency components as the dft approach

numerical approximations to nonlinear conservation laws with locally varying time and space grids an explicit time differencing technique is introduced to approximate nonlinear conservation laws this differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments previous stability requirements i e the cfl condition placed a global bound on the size of the time increments for scalar monotone approximations in one space dimension using this variable step time differencing convergence to the correct physical solution is proven given only a local cfl condition

time asynchronous relative dimension in space method for multi scale problems in fluid dynamics a novel computational method is presented for solving fluid dynamics equations in the multi scale framework when the system size is an important parameter of the governing equations the method tardis is based on a concurrent transformation of the governing equations in space and time and solving the transformed equations on a uniform cartesian grid with the corresponding causality conditions at the grid interfaces for implementation in the framework of tardis the second order cabaret scheme of karabasov and goloviznin 1 is selected for it provides a good combination of numerical accuracy computational efficiency and simplicity of realisation numerical examples are first provided for several isothermal gas dynamics test problems and then for modelling of molecular fluctuations inside a microscopic flow channel and ultrasound wave propagation through a nano scale region of molecular fluctuations

on convergence of a domain decomposition method for a scalar conservation law in this paper we prove convergence of a domain decomposition method for one dimensional scalar conservation laws by dealing carefully with nonconservative terms at the interface of subdomains the method consists of an explicit scheme in some subdomains and an implicit scheme in other subdomains with a numerical flux being the same as the one used in the explicit scheme although such a multidomain algorithm is not strictly conservative the conservation error ce 0 n delta t is equal to mathcal o delta t regardless of the smoothness of the solution finally two test examples are given to validate convergence and the computational efficiency of the present method

a collection of examples where neville elimination outperforms gaussian elimination neville elimination is an elimination procedure alternative to gaussian elimination it is very useful when dealing with totally positive matrices for which nice stability results are known here we include examples most of them test matrices used in matlab which are not totally positive matrices where neville elimination outperforms gaussian elimination

application of implicit explicit high order runge kutta methods to discontinuous galerkin schemes despite the popularity of high order explicit runge kutta erk methods for integrating semi discrete systems of equations erk methods suffer from severe stability based time step restrictions for very stiff problems we implement a discontinuous galerkin finite element method dgfem along with recently introduced high order implicit explicit runge kutta imex rk schemes to overcome geometry induced stiffness in fluid flow problems the imex algorithms solve the non stiff portions of the domain using explicit methods and isolate and solve the more expensive stiff portions using an l stable stiffly accurate explicit singly diagonally implicit runge kutta method esdirk furthermore we apply adaptive time step controllers based on the embedded temporal error predictors we demonstrate in a number of numerical test problems that imex methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid induced stiffness

a parallel local timestepping runge kutta discontinuous galerkin method with applications to coastal ocean modeling abstract geophysical flows over complex domains often encompass both coarse and highly resolved regions approximating these flows using shock capturing methods with explicit timestepping gives rise to a courant friedrichs lewy cfl timestep constraint this approach can result in small global timesteps often dictated by flows in small regions vastly increasing computational effort over the whole domain one approach for coping with this problem is to use locally varying timesteps in previous work we formulated a local timestepping lts method within a runge kutta discontinuous galerkin framework and demonstrated the accuracy and efficiency of this method on serial machines for relatively small scale shallow water applications for more realistic models involving large domains and highly complex physics the lts method must be parallelized for multi core parallel computers furthermore additional physics such as strong wind forcing can effect the choice of local timesteps in this paper we describe a parallel lts method parallelized using domain decomposition and mpi we demonstrate the method on tidal flows and hurricane storm surge applications in the coastal regions of the western north atlantic ocean

an asynchronous scheme with local time stepping for multi scale transport problems application to gas discharges this paper presents an asynchronous integration scheme with local time stepping for transport problems the concept consists in associating refresh time tags to the interface fluxes between cells and to the source terms within the cells rather than to the cell themselves this scheme is less diffusive numerically than its synchronous equivalent this method is very effective in terms of computation time for problems with localized sharp minima in the cfl condition the method is then applied to dielectric barrier discharges for aerodynamic flow control

asynchronous variational integrators we describe a class of asynchronous variational integrators avi for nonlinear elastodynamics the avis are characterized by the following distinguishing attributes i the algorithms permit the selection of independent time steps in each element and the local time steps need not bear an integral relation to each other ii the algorithms derive from a spacetime form of a discrete version of hamilton s principle as a consequence of this variational structure the algorithms conserve local energy and momenta exactly subject to solvability of the local time steps numerical tests reveal that even when local energy balance is not enforced exactly the global and local energy behavior of the avis is quite remarkable a property which can probably be traced to the symplectic nature of the algorithm

optimal numerical parameterization of discontinuous galerkin method applied to wave propagation problems this paper deals with the high order discontinuous galerkin dg method for solving wave propagation problems first we develop a one dimensional dg scheme and numerically compute dissipation and dispersion errors for various polynomial orders an optimal combination of time stepping scheme together with the high order dg spatial scheme is presented it is shown that using a time stepping scheme with the same formal accuracy as the dg scheme is too expensive for the range of wave numbers that is relevant for practical applications an efficient implementation of a high order dg method in three dimensions is presented using 1d convergence results we further show how to adequately choose elementary polynomial orders in order to equi distribute a priori the discretization error we also show a straightforward manner to allow variable polynomial orders in a dg scheme we finally propose some numerical examples in the field of aero acoustics

probabilistic analysis of complex gaussian elimination without pivoting abstract we consider gaussian elimination without pivoting applied to complex gaussian matrices x c n n we first study some independence properties of the elements of the lu factors of x based on this we then derive the probability distributions for all the l and u elements and obtain bounds for the probabilities of the occurrence of small pivots and large growth factors numerical experiments are presented to support the theoretical results and discussions are made to relate the results to the crucial practical problems of numerical stability of ge

moving mesh methods with locally varying time steps the time steps associated with moving mesh methods are proportional to the smallest mesh size in space and as a result they are very small at each time level for some practical problems the physical phenomena develop dynamically singular or nearly singular solutions in fairly localized regions and therefore the smallest time step at each time level occurs only in these localized regions in this work we will develop a local time stepping algorithm for the moving mesh methods the principal idea will be demonstrated by investigating the nonlinear hyperbolic conservation laws numerical experiments are carried out to demonstrate the efficiency and robustness of the proposed methods

multiadaptive galerkin methods for odes iii a priori error estimates the multiadaptive continuous discontinuous galerkin methods mcgq and mdgq for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution in this paper we prove general order a priori error estimates for the mcgq and mdgq methods to prove the error estimates we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution the estimates then follow from interpolation estimates together with stability estimates for the discrete dual solution