In this paper,we propose a coupled discontinuous Galerkin(DG)and continuous Galerkin(CG)scheme for solving the nonlinear evolution Davey-Stewartson(DS)system in dimensionless form.The DS system consists of two coupled...In this paper,we propose a coupled discontinuous Galerkin(DG)and continuous Galerkin(CG)scheme for solving the nonlinear evolution Davey-Stewartson(DS)system in dimensionless form.The DS system consists of two coupled nonlinear and complex structure partial differential equations.The wave’s amplitude in the first equation is solved by the high-efficiency local DG method,and the velocity in the second equation is obtained by a standard CG method.No matching conditions are needed for the two finite element spaces since the normal component of the velocity is continuous across element boundaries.The main strengths of our approach are that we combine the advantage of DG and CG methods,using DG methods handling the nonlinear Schr¨odinger equation to obtain high parallelizability and high-order formal accuracy,using the continuous finite elements solving the velocity to maintain total energy conservation.We prove the energy-conserving properties of our scheme and error estimates in L2-norm.However,the non-linearity terms bring a lot of trouble to the proof of error estimates.With the help of energy-conserving properties,we construct a series of energy equations to obtain error estimates.Numerical tests for different types of systems are presented to clarify the effectiveness of numerical methods.展开更多
The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomia...The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials,modified to accommodate a C~0-continuous expansion. Computationally and theoretically, by increasing the polynomial order p,high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.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
基金supported by the National Natural Science Foundation of China(Grant No.12071455).
文摘In this paper,we propose a coupled discontinuous Galerkin(DG)and continuous Galerkin(CG)scheme for solving the nonlinear evolution Davey-Stewartson(DS)system in dimensionless form.The DS system consists of two coupled nonlinear and complex structure partial differential equations.The wave’s amplitude in the first equation is solved by the high-efficiency local DG method,and the velocity in the second equation is obtained by a standard CG method.No matching conditions are needed for the two finite element spaces since the normal component of the velocity is continuous across element boundaries.The main strengths of our approach are that we combine the advantage of DG and CG methods,using DG methods handling the nonlinear Schr¨odinger equation to obtain high parallelizability and high-order formal accuracy,using the continuous finite elements solving the velocity to maintain total energy conservation.We prove the energy-conserving properties of our scheme and error estimates in L2-norm.However,the non-linearity terms bring a lot of trouble to the proof of error estimates.With the help of energy-conserving properties,we construct a series of energy equations to obtain error estimates.Numerical tests for different types of systems are presented to clarify the effectiveness of numerical methods.
文摘The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials,modified to accommodate a C~0-continuous expansion. Computationally and theoretically, by increasing the polynomial order p,high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.
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.
