High-Order Hybrid WCNS-CPR Scheme for Shock Capturing of Conservation Laws

By introducing hybrid technique into high-order CPR (correction procedure via reconstruction) scheme, a novel hybrid WCNS-CPR scheme is developed for efficient supersonic simulations. Firstly, a shock detector based on nonlinear weights is used to identify grid cells with high gradients or discontinuities throughout the whole flow field. Then, WCNS (weighted compact nonlinear scheme) is adopted to capture shocks in these areas, while the smooth area is calculated by CPR. A strategy to treat the interfaces of the two schemes is developed, which maintains high-order accuracy. Convergent order of accuracy and shock-capturing ability are tested in several numerical experiments; the results of which show that this hybrid scheme achieves expected high-order accuracy and high resolution, is robust in shock capturing, and has less computational cost compared to the WCNS.

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A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

Communications on Applied Mathematics and Computation ◽

10.1007/s42967-021-00120-x ◽

2021 ◽

Author(s):

Johannes Markert ◽

Gregor Gassner ◽

Stefanie Walch

Keyword(s):

Discontinuous Galerkin ◽

Adaptive Mesh Refinement ◽

Adaptive Mesh ◽

High Order ◽

Shock Capturing ◽

High Order Accuracy ◽

Finite Volume Scheme ◽

Galerkin Methods ◽

Order Accuracy ◽

New Strategy

AbstractIn this paper, a new strategy for a sub-element-based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low-to-high-order discretizations on this set of data, including a first-order finite volume scheme up to the full-order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high-order accuracy as possible, even in simulations with very strong shocks, as, e.g., presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

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A HIGH ORDER KINETIC FLUX-SPLITTING METHOD FOR THE SPECIAL RELATIVISTIC HYDRODYNAMICS

International Journal of Computational Methods ◽

10.1142/s0219876205000338 ◽

2005 ◽

Vol 02 (01) ◽

pp. 49-74 ◽

Cited By ~ 2

Author(s):

SHAMSUL QAMAR ◽

GERALD WARNECKE

Keyword(s):

High Resolution ◽

Kinetic Theory ◽

Splitting Method ◽

High Order ◽

Shock Capturing ◽

High Order Accuracy ◽

Order Kinetic ◽

Relativistic Hydrodynamics ◽

Order Accuracy ◽

Flux Splitting

In this article we present a flux splitting method based on gas-kinetic theory for the special relativistic hydrodynamics (SRHD) [Landau and Lifshitz, Fluid Mechanics, Pergamon New York, 1987] in one and two space dimensions. This kinetic method is based on the direct splitting of the macroscopic flux functions with the consideration of particle transport. At the same time, particle "collisions" are implemented in the free transport process to reduce numerical dissipation. Due to the nonlinear relations between conservative and primitive variables and the consequent complexity of the Jacobian matrix, the multi-dimensional shock-capturing numerical schemes for SRHD are computationally more expensive. All the previous methods presented for the solution of these equations were based on the macroscopic continuum description. These upwind high-resolution shock-capturing (HRSC) schemes, which were originally made for non-relativistic flows, were extended to SRHD. However our method, which is based on kinetic theory is more related to the physics of these equations and is very efficient, robust, and easy to implement. In order to get high order accuracy in space, we use a third order central weighted essentially non-oscillatory (CWENO) finite difference interpolation routine. To achieve high order accuracy in time we use a Runge-Kutta time stepping method. The one- and two-dimensional computations reported in this paper show the desired accuracy, high resolution, and robustness of the method.

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