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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.
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 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. These 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.