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
This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin （IPDG） formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Di...This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin （IPDG） formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Dirichlet-to-Neumann （DtN） boundary con- dition. An effective iterative approach is proposed to localize the global DtN boundary condition, which facilitates the implementation of multi-domain methods, and the treat- ment for complex geometry of the scatterers. Under a discontinuous Galerkin formulation, the proposed method allows to use polynomial basis functions of different degree on dif- ferent subdomains, and more importantly, explicit wave number dependence estimates of the spectral scheme can be derived, which is somehow implausible for a multi-domain continuous Galerkin formulation.展开更多
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.
基金The work of the first author was partially supported by the National Natural Science Foundation of China （11026065 and 11101196）. This work was largely done when this author worked as a Research Fellow in Nanyang Technological University. The work of the second author was supported by the National Natural Science Foundation of China （11201166）. The work of the third author was supported by Singapore MOE Tier 1 Grant （2013-2016）, and Singapore A＊STAR-SERC-PSF Grant 122-PSF-0007.
文摘This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin （IPDG） formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Dirichlet-to-Neumann （DtN） boundary con- dition. An effective iterative approach is proposed to localize the global DtN boundary condition, which facilitates the implementation of multi-domain methods, and the treat- ment for complex geometry of the scatterers. Under a discontinuous Galerkin formulation, the proposed method allows to use polynomial basis functions of different degree on dif- ferent subdomains, and more importantly, explicit wave number dependence estimates of the spectral scheme can be derived, which is somehow implausible for a multi-domain continuous Galerkin formulation.
