The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework i...The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework is the automatic way of getting arbitrarily high order methods,which can be put in the Runge-Kutta(RK)form.The drawback is the larger computational cost with respect to the most used RK methods.To reduce such cost,in an explicit setting,we propose an efcient modifcation:we introduce interpolation processes between the DeC iterations,decreasing the computational cost associated to the low order ones.We provide the Butcher tableaux of the new modifed methods and we study their stability,showing that in some cases the computational advantage does not afect the stability.The fexibility of the novel modifcation allows nontrivial applications to PDEs and construction of adaptive methods.The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.展开更多
In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Re...In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Restarted Adomian Decomposition Method(RADM)for solving the two equations of different types involving dissimilar boundary and nonlocal conditions.The algorithm presented transforms the given nonlocal initial boundary value problem to a local Dirichlet one and then employs the RADM for the numerical treatment.Numerical comparisons were made between our proposed method and the Adomian Decomposition Method(ADM)to demonstrate the efficiency and performance of the proposed method.展开更多
Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of ...Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.展开更多
The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the un...The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.展开更多
High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of th...High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.展开更多
Dear Editor,This letter focuses on the distributed cooperative regulation problem for a class of networked re-entrant manufacturing systems(RMSs).The networked system is structured with a three-tier architecture:the p...Dear Editor,This letter focuses on the distributed cooperative regulation problem for a class of networked re-entrant manufacturing systems(RMSs).The networked system is structured with a three-tier architecture:the production line,the manufacturing layer and the workshop layer.The dynamics of re-entrant production lines are governed by hyperbolic partial differential equations(PDEs)based on the law of mass conservation.展开更多
Higher order finite difference Weighted Essentially Non-oscillatory(WENO)schemes for conservation laws represent a technology that has been reasonably consolidated.They are extremely popular because,when applied to mu...Higher order finite difference Weighted Essentially Non-oscillatory(WENO)schemes for conservation laws represent a technology that has been reasonably consolidated.They are extremely popular because,when applied to multidimensional problems,they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes.They come in two flavors.There is the classical finite difference WENO(FD-WENO)method(Shu and Osher in J.Comput.Phys.83:32–78,1989).However,in recent years there is also an alternative finite difference WENO(AFD-WENO)method which has recently been formalized into a very useful general-purpose algorithm for conservation laws(Balsara et al.in Efficient alternative finite difference WENO schemes for hyperbolic conservation laws,submitted to CAMC,2023).However,the FD-WENO algorithm has only very recently been formulated for hyperbolic systems with non-conservative products(Balsara et al.in Efficient finite difference WENO scheme for hyperbolic systems with non-conservative products,to appear CAMC,2023).In this paper,we show that there are substantial advantages in obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative products.Such an algorithm is documented in this paper.We present an AFD-WENO formulation in a fluctuation form that is carefully engineered to retrieve the flux form when that is warranted and nevertheless extends to non-conservative products.The method is flexible because it allows any Riemann solver to be used.The formulation we arrive at is such that when non-conservative products are absent it reverts exactly to the formulation in the second citation above which is in the exact flux conservation form.The ability to transition to a precise conservation form when non-conservative products are absent ensures,via the Lax-Wendroff theorem,that shock locations will be exactly captured by the method.We present two formulations of AFD-WENO that can be used with hyperbolic systems with non-conservative products and stiff source terms with slightly differing computational complexities.The speeds of our new AFD-WENO schemes are compared to the speed of the classical FD-WENO algorithm from the first of the above-cited papers.At all orders,AFD-WENO outperforms FD-WENO.We also show a very desirable result that higher order variants of AFD-WENO schemes do not cost that much more than their lower order variants.This is because the larger number of floating point operations associated with larger stencils is almost very efficiently amortized by the CPU when the AFD-WENO code is designed to be cache friendly.This should have great,and very beneficial,implications for the role of our AFD-WENO schemes in the Peta-and Exascale computing.We apply the method to several stringent test problems drawn from the Baer-Nunziato system,two-layer shallow water equations,and the multicomponent debris flow.The method meets its design accuracy for the smooth flow and can handle stringent problems in one and multiple dimensions.Because of the pointwise nature of its update,AFD-WENO for hyperbolic systems with non-conservative products is also shown to be a very efficient performer on problems with stiff source terms.展开更多
Higher order finite difference Weighted Essentially Non-Oscillatory(FD-WENO)schemes for conservation laws are extremely popular because,for multidimensional problems,they offer high order accuracy at a fraction of the...Higher order finite difference Weighted Essentially Non-Oscillatory(FD-WENO)schemes for conservation laws are extremely popular because,for multidimensional problems,they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes.Such schemes come in two formulations.The very popular classical FD-WENO method(Shu and Osher J Comput Phys 83:32–78,1989)relies on two reconstruction steps applied to two split fluxes.However,the method cannot accommodate different types of Riemann solvers and cannot preserve free stream boundary conditions on curvilinear meshes.This limits its utility.The alternative FD-WENO(AFD-WENO)method can overcome these deficiencies,however,much less work has been done on this method.The reasons are three-fold.First,it is difficult for the casual reader to understand the intricate logic that requires higher order derivatives of the fluxes to be evaluated at zone boundaries.The analytical methods for deriving the update equation for AFD-WENO schemes are somewhat recondite.To overcome that difficulty,we provide an easily accessible script that is based on a computer algebra system in Appendix A of this paper.Second,the method relies on interpolation rather than reconstruction,and WENO interpolation formulae have not been documented in the literature as thoroughly as WENO reconstruction formulae.In this paper,we explicitly provide all necessary WENO interpolation formulae that are needed for implementing the AFD-WENO up to the ninth order.The third reason is that the AFD-WENO requires higher order derivatives of the fluxes to be available at zone boundaries.Since those derivatives are usually obtained by finite differencing the zone-centered fluxes,they become susceptible to a Gibbs phenomenon when the solution is non-smooth.The inclusion of those fluxes is also crucially important for preserving the order property when the solution is smooth.This has limited the utility of the AFD-WENO in the past even though the method per se has many desirable features.Some efforts to mitigate the effect of finite differencing of the fluxes have been tried,but so far they have been done on a case by case basis for the PDE being considered.In this paper we find a general-purpose strategy that is based on a different type of the WENO interpolation.This new WENO interpolation takes the first derivatives of the fluxes at zone centers as its inputs and returns the requisite non-linearly hybridized higher order derivatives of flux-like terms at the zone boundaries as its output.With these three advances,we find that the AFD-WENO becomes a robust and general-purpose solution strategy for large classes of conservation laws.It allows any Riemann solver to be used.The AFD-WENO has a computational complexity that is entirely comparable to the classical FD-WENO,because it relies on two interpolation steps which cost the same as the two reconstruction steps in the classical FD-WENO.We apply the method to several stringent test problems drawn from Euler flow,relativistic hydrodynamics(RHD),and ten-moment equations.The method meets its design accuracy for smooth flow and can handle stringent problems in one and multiple dimensions.展开更多
In this paper,we investigate the coupling of the Multi-dimensional Optimal Order Detection(MOOD)method and the Arbitrary high order DERivatives(ADER)approach in order to design a new high order accurate,robust and com...In this paper,we investigate the coupling of the Multi-dimensional Optimal Order Detection(MOOD)method and the Arbitrary high order DERivatives(ADER)approach in order to design a new high order accurate,robust and computationally efficient Finite Volume(FV)scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions,respectively.The Multi-dimensional Optimal Order Detection(MOOD)method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes.It is an arbitrary high-order accurate Finite Volume scheme in space,using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities.In the following work,the time discretization is performed with an elegant and efficient one-step ADER procedure.Doing so,we retain the good properties of the MOOD scheme,that is to say the optimal high-order of accuracy is reached on smooth solutions,while spurious oscillations near singularities are prevented.The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D,but it also increases the stability of the overall scheme.A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost,robustness,accuracy and efficiency.The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive(memory and/or CPU time),or because it is more accurate for a given grid resolution.A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws:the Euler equations of compressible gas dynamics,the classical equations of ideal magneto-Hydrodynamics(MHD)and finally the relativistic MHD equations(RMHD),which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation.All tests are run on genuinely unstructured grids composed of simplex elements.展开更多
文摘The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework is the automatic way of getting arbitrarily high order methods,which can be put in the Runge-Kutta(RK)form.The drawback is the larger computational cost with respect to the most used RK methods.To reduce such cost,in an explicit setting,we propose an efcient modifcation:we introduce interpolation processes between the DeC iterations,decreasing the computational cost associated to the low order ones.We provide the Butcher tableaux of the new modifed methods and we study their stability,showing that in some cases the computational advantage does not afect the stability.The fexibility of the novel modifcation allows nontrivial applications to PDEs and construction of adaptive methods.The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.
文摘In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Restarted Adomian Decomposition Method(RADM)for solving the two equations of different types involving dissimilar boundary and nonlocal conditions.The algorithm presented transforms the given nonlocal initial boundary value problem to a local Dirichlet one and then employs the RADM for the numerical treatment.Numerical comparisons were made between our proposed method and the Adomian Decomposition Method(ADM)to demonstrate the efficiency and performance of the proposed method.
基金support via NSF grants NSF-19-04774,NSF-AST-2009776,NASA-2020-1241NASA grant 80NSSC22K0628.DSB+3 种基金HK acknowledge support from a Vajra award,VJR/2018/00129a travel grant from Notre Dame Internationalsupport via AFOSR grant FA9550-20-1-0055NSF grant DMS-2010107.
文摘Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.
基金the National Natural Science Foundation of China(Nos.11671282 and 12171339)。
文摘The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.
文摘High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.
文摘Dear Editor,This letter focuses on the distributed cooperative regulation problem for a class of networked re-entrant manufacturing systems(RMSs).The networked system is structured with a three-tier architecture:the production line,the manufacturing layer and the workshop layer.The dynamics of re-entrant production lines are governed by hyperbolic partial differential equations(PDEs)based on the law of mass conservation.
基金support via NSF grant NSF-AST-2009776,NASA grant NASA-2020-1241 and NASA grant 80NSSC22K0628support from a Vajra award,VJR/2018/00129 and also a travel grant from Notre Dame International.CWS acknowledges support via NSF grant DMS-2309249+2 种基金support via the NSF Grants NSF-19-04774,NSF-AST-2009776,NASA-2020-1241,and NASA-80NSSC22K0628support from a Vajra award,VJR/2018/00129support via AFOSR Grant FA9550-20-1-0055 and NSF Grant DMS-2010107.
文摘Higher order finite difference Weighted Essentially Non-oscillatory(WENO)schemes for conservation laws represent a technology that has been reasonably consolidated.They are extremely popular because,when applied to multidimensional problems,they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes.They come in two flavors.There is the classical finite difference WENO(FD-WENO)method(Shu and Osher in J.Comput.Phys.83:32–78,1989).However,in recent years there is also an alternative finite difference WENO(AFD-WENO)method which has recently been formalized into a very useful general-purpose algorithm for conservation laws(Balsara et al.in Efficient alternative finite difference WENO schemes for hyperbolic conservation laws,submitted to CAMC,2023).However,the FD-WENO algorithm has only very recently been formulated for hyperbolic systems with non-conservative products(Balsara et al.in Efficient finite difference WENO scheme for hyperbolic systems with non-conservative products,to appear CAMC,2023).In this paper,we show that there are substantial advantages in obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative products.Such an algorithm is documented in this paper.We present an AFD-WENO formulation in a fluctuation form that is carefully engineered to retrieve the flux form when that is warranted and nevertheless extends to non-conservative products.The method is flexible because it allows any Riemann solver to be used.The formulation we arrive at is such that when non-conservative products are absent it reverts exactly to the formulation in the second citation above which is in the exact flux conservation form.The ability to transition to a precise conservation form when non-conservative products are absent ensures,via the Lax-Wendroff theorem,that shock locations will be exactly captured by the method.We present two formulations of AFD-WENO that can be used with hyperbolic systems with non-conservative products and stiff source terms with slightly differing computational complexities.The speeds of our new AFD-WENO schemes are compared to the speed of the classical FD-WENO algorithm from the first of the above-cited papers.At all orders,AFD-WENO outperforms FD-WENO.We also show a very desirable result that higher order variants of AFD-WENO schemes do not cost that much more than their lower order variants.This is because the larger number of floating point operations associated with larger stencils is almost very efficiently amortized by the CPU when the AFD-WENO code is designed to be cache friendly.This should have great,and very beneficial,implications for the role of our AFD-WENO schemes in the Peta-and Exascale computing.We apply the method to several stringent test problems drawn from the Baer-Nunziato system,two-layer shallow water equations,and the multicomponent debris flow.The method meets its design accuracy for the smooth flow and can handle stringent problems in one and multiple dimensions.Because of the pointwise nature of its update,AFD-WENO for hyperbolic systems with non-conservative products is also shown to be a very efficient performer on problems with stiff source terms.
基金support via the NSF grants NSF-19-04774,NSF-AST-2009776,NASA-2020-1241,and(NASA-80NSSC22K0628)support from a Vajra award(VJR/2018/00129)support via the NSF grant DMS-2309249.
文摘Higher order finite difference Weighted Essentially Non-Oscillatory(FD-WENO)schemes for conservation laws are extremely popular because,for multidimensional problems,they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes.Such schemes come in two formulations.The very popular classical FD-WENO method(Shu and Osher J Comput Phys 83:32–78,1989)relies on two reconstruction steps applied to two split fluxes.However,the method cannot accommodate different types of Riemann solvers and cannot preserve free stream boundary conditions on curvilinear meshes.This limits its utility.The alternative FD-WENO(AFD-WENO)method can overcome these deficiencies,however,much less work has been done on this method.The reasons are three-fold.First,it is difficult for the casual reader to understand the intricate logic that requires higher order derivatives of the fluxes to be evaluated at zone boundaries.The analytical methods for deriving the update equation for AFD-WENO schemes are somewhat recondite.To overcome that difficulty,we provide an easily accessible script that is based on a computer algebra system in Appendix A of this paper.Second,the method relies on interpolation rather than reconstruction,and WENO interpolation formulae have not been documented in the literature as thoroughly as WENO reconstruction formulae.In this paper,we explicitly provide all necessary WENO interpolation formulae that are needed for implementing the AFD-WENO up to the ninth order.The third reason is that the AFD-WENO requires higher order derivatives of the fluxes to be available at zone boundaries.Since those derivatives are usually obtained by finite differencing the zone-centered fluxes,they become susceptible to a Gibbs phenomenon when the solution is non-smooth.The inclusion of those fluxes is also crucially important for preserving the order property when the solution is smooth.This has limited the utility of the AFD-WENO in the past even though the method per se has many desirable features.Some efforts to mitigate the effect of finite differencing of the fluxes have been tried,but so far they have been done on a case by case basis for the PDE being considered.In this paper we find a general-purpose strategy that is based on a different type of the WENO interpolation.This new WENO interpolation takes the first derivatives of the fluxes at zone centers as its inputs and returns the requisite non-linearly hybridized higher order derivatives of flux-like terms at the zone boundaries as its output.With these three advances,we find that the AFD-WENO becomes a robust and general-purpose solution strategy for large classes of conservation laws.It allows any Riemann solver to be used.The AFD-WENO has a computational complexity that is entirely comparable to the classical FD-WENO,because it relies on two interpolation steps which cost the same as the two reconstruction steps in the classical FD-WENO.We apply the method to several stringent test problems drawn from Euler flow,relativistic hydrodynamics(RHD),and ten-moment equations.The method meets its design accuracy for smooth flow and can handle stringent problems in one and multiple dimensions.
基金the European Research Council(ERC)under the European Union’s Seventh Framework Programme(FP7/2007-2013)the research project STiMulUs,ERC Grant agreement no.278267+1 种基金.R.L.has been partially funded by the ANR under the JCJC project“ALE INC(ubator)3D”the reference LA-UR-13-28795.The authors would like to acknowledge PRACE for awarding access to the SuperMUC supercomputer based in Munich,Germany at the Leibniz Rechenzentrum(LRZ)。
文摘In this paper,we investigate the coupling of the Multi-dimensional Optimal Order Detection(MOOD)method and the Arbitrary high order DERivatives(ADER)approach in order to design a new high order accurate,robust and computationally efficient Finite Volume(FV)scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions,respectively.The Multi-dimensional Optimal Order Detection(MOOD)method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes.It is an arbitrary high-order accurate Finite Volume scheme in space,using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities.In the following work,the time discretization is performed with an elegant and efficient one-step ADER procedure.Doing so,we retain the good properties of the MOOD scheme,that is to say the optimal high-order of accuracy is reached on smooth solutions,while spurious oscillations near singularities are prevented.The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D,but it also increases the stability of the overall scheme.A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost,robustness,accuracy and efficiency.The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive(memory and/or CPU time),or because it is more accurate for a given grid resolution.A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws:the Euler equations of compressible gas dynamics,the classical equations of ideal magneto-Hydrodynamics(MHD)and finally the relativistic MHD equations(RMHD),which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation.All tests are run on genuinely unstructured grids composed of simplex elements.
