Abstract
The Arbitrary accuracy Derivatives Riemann problem method (ADER) scheme is a new high order numerical scheme based on the concept of finite volume integration, and it is very easy to be extended up to any order of space and time accuracy by using a Taylor time expansion at the cell interface position. So far the approach has been applied successfully to flow mechanics problems. Our objective here is to carry out the extension of multidimensional ADER schemes to multidimensional MHD systems of conservation laws by calculating several MHD problems in one and two dimensions: (ⅰ) Brio-Wu shock tube problem, (ⅱ) Dai-Woodward shock tube problem, (ⅲ) Orszag-Tang MHD vortex problem. The numerical results prove that the ADER scheme possesses the ability to solve MHD problem, remains high order accuracy both in space and time, keeps precise in capturing the shock. Meanwhile, the compared tests show that the ADER scheme can restrain the oscillation and obtain the high order non-oscillatory result.
Highlights
The classical Riemann problem is the Cauchy problem for a system of hyperbolic conservation laws, with initial condition consisting of two constant states separated by a discontinuity
According to Toro’s concept of the Generalized Riemann problem[5], the initial condition consists of two kth order polynomial functions, which are denoted as the corresponding Generalized Riemann problem by GRPk
ZHANG Yanyan et al.: Application of accuracy Derivatives Riemann problem method (ADER) Scheme in MHD Simulation higher-order spatial derivatives of the initial condition for the GRP away from the origin vanish identically, which corresponds to the classical piece-wise constant data Riemann problem
Summary
The classical Riemann problem is the Cauchy problem for a system of hyperbolic conservation laws, with initial condition consisting of two constant states separated by a discontinuity. ZHANG Yanyan et al.: Application of ADER Scheme in MHD Simulation higher-order spatial derivatives of the initial condition for the GRP away from the origin vanish identically, which corresponds to the classical piece-wise constant data Riemann problem. The Arbitrary accuracy Derivatives Riemann problem method (ADER) approach can be regarded as a further development of the MGRP scheme in that it breaks the barrier of second-order accuracy and allows the construction of arbitrarily high-order accurate schemes, both in time and space. This approach was put forward firstly by Toro and his collaborators[7], where they achieved 10th order of accuracy in both space and time. The numerical results show that the ADER scheme possesses the ability to solve MHD supported by problem, and remains high order accuracy both in space and time
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