In this paper,we present a novel spatial reconstruction scheme,called AENO,that results from a special averaging of the ENO polynomial and its closest neighbour,while retaining the stencil direction decided by the ENO...In this paper,we present a novel spatial reconstruction scheme,called AENO,that results from a special averaging of the ENO polynomial and its closest neighbour,while retaining the stencil direction decided by the ENO choice.A variant of the scheme,called m-AENO,results from averaging the modified ENO(m-ENO)polynomial and its closest neighbour.The concept is thoroughly assessed for the one-dimensional linear advection equation and for a one-dimensional non-linear hyperbolic system,in conjunction with the fully discrete,high-order ADER approach implemented up to fifth order of accuracy in both space and time.The results,as compared to the conventional ENO,m-ENO and WENO schemes,are very encouraging.Surprisingly,our results show that the L_(1)-errors of the novel AENO approach are the smallest for most cases considered.Crucially,for a chosen error size,AENO turns out to be the most efficient method of all five methods tested.展开更多
The Arbitrary accuracy Derivatives Riemann problem method(ADER) scheme is a new high order numerical scheme based on the concept of finite volume integration,and it is very easy to be extended up to any order of space...The Arbitrary accuracy Derivatives Riemann problem method(ADER) scheme is a new high order numerical scheme based on the concept of finite volume integration,and it is very easy to be extended up to any order of space and time accuracy by using a Taylor time expansion at the cell interface position.So far the approach has been applied successfully to flow mechanics problems.Our objective here is to carry out the extension of multidimensional ADER schemes to multidimensional MHD systems of conservation laws by calculating several MHD problems in one and two dimensions: (ⅰ) Brio-Wu shock tube problem,(ⅱ) Dai-Woodward shock tube problem,(ⅲ) Orszag-Tang MHD vortex problem.The numerical results prove that the ADER scheme possesses the ability to solve MHD problem,remains high order accuracy both in space and time,keeps precise in capturing the shock.Meanwhile,the compared tests show that the ADER scheme can restrain the oscillation and obtain the high order non-oscillatory result.展开更多
The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years,including both methodological aspects as well as very ambitious applications.In spit...The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years,including both methodological aspects as well as very ambitious applications.In spite of methodological progress,the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem(GRP)remain the centre of attention in the ADER approach.In the original formulation of ADER schemes,the proposed solution procedure for the GRP was based on(i)Taylor series expansion of the solution in time right at the element interface,(ii)subsequent application of the Cauchy-Kowalewskaya procedure to convert time derivatives to functionals of space derivatives,and(iii)solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series expansion.For realistic problems the Cauchy-Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly desirable.In this paper we propose a new class of solvers for the GRP that avoid the Cauchy-Kowalewskaya procedure and result in simpler ADER schemes.This is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element interface.Still relying on a time Taylor series expansion of the solution at the interface,the time derivatives are then easily calculated from the time-reconstruction polynomial.The resulting schemes are called ADER-TR.A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method,thus deducing linear stability regions.Also,via careful numerical experiments,we deduce stability regions for the corresponding non-linear schemes.Numerical examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER methods.This paper is restricted to the one-dimensional scalar case with source term,but preliminary results for the one-dimensional Euler equations indicate that the time-reconstruction approach offers significant advantages not only in terms of ease of implementation but also in terms of efficiency for the high-order range schemes.展开更多
We construct an approximate Riemann solver for scalar advection-diffusion equations with piecewise polynomial initial data.The objective is to handle advection and diffusion simultaneously to reduce the inherent numer...We construct an approximate Riemann solver for scalar advection-diffusion equations with piecewise polynomial initial data.The objective is to handle advection and diffusion simultaneously to reduce the inherent numerical diffusion produced by the usual advection flux calculations.The approximate solution is based on the weak formulation of the Riemann problem and is solved within a space-time discontinuous Galerkin approach with two subregions.The novel generalized Riemann solver produces piecewise polynomial solutions of the Riemann problem.In conjunction with a recovery polynomial,the Riemann solver is then applied to define the numerical flux within a finite volume method.Numerical results for a piecewise linear and a piecewise parabolic approximation are shown.These results indicate a reduction in numerical dissipation compared with the conventional separated flux calculation of advection and diffusion.Also,it is shown that using the proposed solver only in the vicinity of discontinuities gives way to an accurate and efficient finite volume scheme.展开更多
We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretiz...We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretization, while the time integration is performed at the same order of accuracy thanks to an Arbitrary high order DERivatives (ADER) method. The orders of convergence of the three ADER-IPDG methods are carefully examined through numerical illustrations, showing that the approach is consistent, accurate, and efficient. The numerical results indicate that the symmetric version of IPDG is typically more accurate and more efficient compared to the other approaches.展开更多
文摘In this paper,we present a novel spatial reconstruction scheme,called AENO,that results from a special averaging of the ENO polynomial and its closest neighbour,while retaining the stencil direction decided by the ENO choice.A variant of the scheme,called m-AENO,results from averaging the modified ENO(m-ENO)polynomial and its closest neighbour.The concept is thoroughly assessed for the one-dimensional linear advection equation and for a one-dimensional non-linear hyperbolic system,in conjunction with the fully discrete,high-order ADER approach implemented up to fifth order of accuracy in both space and time.The results,as compared to the conventional ENO,m-ENO and WENO schemes,are very encouraging.Surprisingly,our results show that the L_(1)-errors of the novel AENO approach are the smallest for most cases considered.Crucially,for a chosen error size,AENO turns out to be the most efficient method of all five methods tested.
基金Supported by the National Natural Science Foundation of China(40904050,40874077)the Specialized Research Fund for State Key Laboratories
文摘The Arbitrary accuracy Derivatives Riemann problem method(ADER) scheme is a new high order numerical scheme based on the concept of finite volume integration,and it is very easy to be extended up to any order of space and time accuracy by using a Taylor time expansion at the cell interface position.So far the approach has been applied successfully to flow mechanics problems.Our objective here is to carry out the extension of multidimensional ADER schemes to multidimensional MHD systems of conservation laws by calculating several MHD problems in one and two dimensions: (ⅰ) Brio-Wu shock tube problem,(ⅱ) Dai-Woodward shock tube problem,(ⅲ) Orszag-Tang MHD vortex problem.The numerical results prove that the ADER scheme possesses the ability to solve MHD problem,remains high order accuracy both in space and time,keeps precise in capturing the shock.Meanwhile,the compared tests show that the ADER scheme can restrain the oscillation and obtain the high order non-oscillatory result.
基金G.I.Montecinos thanks the National Chilean Fund for Scientific and Technological Development,FONDECYT(Fondo Nacional de Desarrollo Científico y Tecnológico),in the frame of the project for Initiation in Research 11180926
文摘The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years,including both methodological aspects as well as very ambitious applications.In spite of methodological progress,the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem(GRP)remain the centre of attention in the ADER approach.In the original formulation of ADER schemes,the proposed solution procedure for the GRP was based on(i)Taylor series expansion of the solution in time right at the element interface,(ii)subsequent application of the Cauchy-Kowalewskaya procedure to convert time derivatives to functionals of space derivatives,and(iii)solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series expansion.For realistic problems the Cauchy-Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly desirable.In this paper we propose a new class of solvers for the GRP that avoid the Cauchy-Kowalewskaya procedure and result in simpler ADER schemes.This is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element interface.Still relying on a time Taylor series expansion of the solution at the interface,the time derivatives are then easily calculated from the time-reconstruction polynomial.The resulting schemes are called ADER-TR.A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method,thus deducing linear stability regions.Also,via careful numerical experiments,we deduce stability regions for the corresponding non-linear schemes.Numerical examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER methods.This paper is restricted to the one-dimensional scalar case with source term,but preliminary results for the one-dimensional Euler equations indicate that the time-reconstruction approach offers significant advantages not only in terms of ease of implementation but also in terms of efficiency for the high-order range schemes.
基金This work was supported by the German Research Foundation(DFG)through the Collaborative Research Center SFB TRR 75 Droplet Dynamics Under Extreme Ambient Conditions
文摘We construct an approximate Riemann solver for scalar advection-diffusion equations with piecewise polynomial initial data.The objective is to handle advection and diffusion simultaneously to reduce the inherent numerical diffusion produced by the usual advection flux calculations.The approximate solution is based on the weak formulation of the Riemann problem and is solved within a space-time discontinuous Galerkin approach with two subregions.The novel generalized Riemann solver produces piecewise polynomial solutions of the Riemann problem.In conjunction with a recovery polynomial,the Riemann solver is then applied to define the numerical flux within a finite volume method.Numerical results for a piecewise linear and a piecewise parabolic approximation are shown.These results indicate a reduction in numerical dissipation compared with the conventional separated flux calculation of advection and diffusion.Also,it is shown that using the proposed solver only in the vicinity of discontinuities gives way to an accurate and efficient finite volume scheme.
文摘We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretization, while the time integration is performed at the same order of accuracy thanks to an Arbitrary high order DERivatives (ADER) method. The orders of convergence of the three ADER-IPDG methods are carefully examined through numerical illustrations, showing that the approach is consistent, accurate, and efficient. The numerical results indicate that the symmetric version of IPDG is typically more accurate and more efficient compared to the other approaches.
