A two-stage fourth order time-accurate discretization for Lax--Wendroff type flow solvers II. High order numerical boundary conditions
| Autor: | Jiequan Li, Zhifang Du |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Physics and Astronomy (miscellaneous)
Discretization Lax–Wendroff method 010103 numerical & computational mathematics 01 natural sciences 010305 fluids & plasmas symbols.namesake 0103 physical sciences FOS: Mathematics Applied mathematics Boundary value problem Mathematics - Numerical Analysis 0101 mathematics Boundary cell Mathematics Numerical Analysis Partial differential equation Finite volume method Applied Mathematics Numerical Analysis (math.NA) Computer Science Applications Computational Mathematics Modeling and Simulation Jacobian matrix and determinant symbols 65M08 76M12 35L60 35L65 76N15 Interpolation |
| DOI: | 10.48550/arxiv.1801.00990 |
| Popis: | This paper serves to treat boundary conditions numerically with high order accuracy in order to match the two-stage fourth-order finite volume schemes for hyperbolic problems developed in [{\em J. Li and Z. Du, A two-stage fourth order time-accurate discretization {L}ax--{W}endroff type flow solvers, {I}. {H}yperbolic conservation laws, SIAM, J. Sci. Comput., 38 (2016), pp.~A3046--A3069}]. As such, it is significant when capturing small scale structures near physical boundaries. Different from previous contributions in literature, the current approach constructs a fourth order accurate approximation to boundary conditions by only using the Jacobian of the flux function (characteristic information) instead of its successive differentiation leading to tensors of high ranks in the inverse Lax-Wendroff method. Technically, data in several ghost cells are constructed with interpolation so that the interior scheme can be implemented over boundary cells, and theoretical boundary condition has to be modified properly at intermediate stages so as to make the two-stage scheme over boundary cells fully consistent with that over interior cells. This highlights the fact that {\em continuous boundary conditions only match continuous partial differential equations (PDEs), and they must be approximated in a consistent way (even though it could be exactly valued) when the PDEs are discretized.} Several numerical examples are provided to illustrate the performance of the current approach when dealing with general boundary conditions. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|