In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state pr...In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.展开更多
In this paper,we present a Runge-Kutta Discontinuous Galerkin(RKDG)method for solving the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations under the Lagrangian framework.The fluid part of the idea...In this paper,we present a Runge-Kutta Discontinuous Galerkin(RKDG)method for solving the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations under the Lagrangian framework.The fluid part of the ideal MHD equations along with z-component of the magnetic induction equation are discretized using a DG method based on linear Taylor expansions.By using the magnetic fluxfreezing principle which is the integral form of the magnetic induction equation of the ideal MHD,an exactly divergence-free numerical magnetic field can be obtained.The nodal velocities and the corresponding numerical fluxes are explicitly calculated by solving multidirectional approximate Riemann problems.Two kinds of limiter are proposed to inhibit the non-physical oscillation around the shock wave,and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the rotor problems.This Lagrangian RKDG method conserves mass,momentum,and total energy.Several numerical tests are presented to demonstrate the accuracy and robustness of the proposed scheme.展开更多
In this paper,Runge-Kutta Discontinuous Galerkin（RKDG） finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discreti...In this paper,Runge-Kutta Discontinuous Galerkin（RKDG） finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ？ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ？ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.展开更多
We propose a new two-dimensional blood flow reduced model taking into account of complex artery geometry as in the case of severe aneurysm.We derive the model from the three-dimensional Navier-Stokes equations written...We propose a new two-dimensional blood flow reduced model taking into account of complex artery geometry as in the case of severe aneurysm.We derive the model from the three-dimensional Navier-Stokes equations written in a curvilinear coordinate system under the thin-artery assumption,with boundary conditions including wall tissue deformation.We show that the model is energetically consistent with the full Navier-Stokes problem.This model,obtained via radial averaging,is,up to our knowledge,the first one.It has the advantage of being more accurate than the classical one-dimensional models and being solved in a reasonable time in comparison with the Navier-Stokes models.To this purpose,we use a Runge-Kutta Discontinuous Galerkin(RKDG)method to solve the two-dimensional problem.We end the paper with several numerical test cases to show the efficiency and robustness of the numerical model,and in particular,we show the limit of the one-dimensional models in the case of a severe aneurysm.展开更多
基金supported by the NSFC Grant No.11872210 and Grant No.MCMS-I-0120G01Chi-Wang Shu:Research is supported by the AFOSR Grant FA9550-20-1-0055 and the NSF Grant DMS-2010107Jianxian Qiu:Research is supported by the NSFC Grant No.12071392.
文摘In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.
基金supported by National Natural Science Foundation of China(12071046,11671049,91330107,11571002 and 11702028)China Postdoctoral Science Foundation(2020TQ0013).
文摘In this paper,we present a Runge-Kutta Discontinuous Galerkin(RKDG)method for solving the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations under the Lagrangian framework.The fluid part of the ideal MHD equations along with z-component of the magnetic induction equation are discretized using a DG method based on linear Taylor expansions.By using the magnetic fluxfreezing principle which is the integral form of the magnetic induction equation of the ideal MHD,an exactly divergence-free numerical magnetic field can be obtained.The nodal velocities and the corresponding numerical fluxes are explicitly calculated by solving multidirectional approximate Riemann problems.Two kinds of limiter are proposed to inhibit the non-physical oscillation around the shock wave,and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the rotor problems.This Lagrangian RKDG method conserves mass,momentum,and total energy.Several numerical tests are presented to demonstrate the accuracy and robustness of the proposed scheme.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11261035,11171038,and 10771019)the Science Reaearch Foundation of Institute of Higher Education of Inner Mongolia Autonomous Region,China (Grant No. NJZZ12198)the Natural Science Foundation of Inner Mongolia Autonomous Region,China (Grant No. 2012MS0102)
文摘In this paper,Runge-Kutta Discontinuous Galerkin（RKDG） finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ？ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ？ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.
文摘We propose a new two-dimensional blood flow reduced model taking into account of complex artery geometry as in the case of severe aneurysm.We derive the model from the three-dimensional Navier-Stokes equations written in a curvilinear coordinate system under the thin-artery assumption,with boundary conditions including wall tissue deformation.We show that the model is energetically consistent with the full Navier-Stokes problem.This model,obtained via radial averaging,is,up to our knowledge,the first one.It has the advantage of being more accurate than the classical one-dimensional models and being solved in a reasonable time in comparison with the Navier-Stokes models.To this purpose,we use a Runge-Kutta Discontinuous Galerkin(RKDG)method to solve the two-dimensional problem.We end the paper with several numerical test cases to show the efficiency and robustness of the numerical model,and in particular,we show the limit of the one-dimensional models in the case of a severe aneurysm.
