<p>This paper is devoted to the description and validation of a new implementation of a fourth-order space–time conservation-element and solution-element (CESE) scheme to numerically solve the time-dependent, three-dimensional (3D) magnetohydrodynamic (MHD) equations. The core of the scheme is that, with the aid of a grid staggered in space and time, the conservative variables are advanced by integration of the controlling equation in the space–time four-dimensional domain by utilizing Taylor expansion, and their spatial derivatives are computed by finite difference with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2452_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> <EquationSource Format="TEX">$p$</EquationSource> </InlineEquation> order derivatives from <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2452_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>−</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$p-1$</EquationSource> </InlineEquation> order ones. The new scheme achieves fourth-order accuracy in both space and time simultaneously, using a compact stencil identical to that in the second-order CESE scheme. We provide a general framework for convenience of programming such that the scheme can be easily extended to arbitrarily higher order by including higher-order terms in the Taylor series. A suite of 3D MHD tests demonstrate that the fourth-order CESE scheme at relatively low grid resolutions can obtain reliable solution comparable to the second-order CESE scheme at four-times higher resolution, and showing a very high efficiency in computing by using only around <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2452_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>5</mn> <mi mathvariant="normal">%</mi> </math></EquationSource> <EquationSource Format="TEX">$5\%$</EquationSource> </InlineEquation> of the computing resources.</p>

A New Implementation of a Fourth-Order CESE Scheme for 3D MHD Simulations

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发表日期：

2025年3月28日

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Chaowei Jiang,
Ling Zhang

摘要

This paper is devoted to the description and validation of a new implementation of a fourth-order space–time conservation-element and solution-element (CESE) scheme to numerically solve the time-dependent, three-dimensional (3D) magnetohydrodynamic (MHD) equations. The core of the scheme is that, with the aid of a grid staggered in space and time, the conservative variables are advanced by integration of the controlling equation in the space–time four-dimensional domain by utilizing Taylor expansion, and their spatial derivatives are computed by finite difference with $p$ $p$ order derivatives from $p - 1$ $p-1$ order ones. The new scheme achieves fourth-order accuracy in both space and time simultaneously, using a compact stencil identical to that in the second-order CESE scheme. We provide a general framework for convenience of programming such that the scheme can be easily extended to arbitrarily higher order by including higher-order terms in the Taylor series. A suite of 3D MHD tests demonstrate that the fourth-order CESE scheme at relatively low grid resolutions can obtain reliable solution comparable to the second-order CESE scheme at four-times higher resolution, and showing a very high efficiency in computing by using only around $5 %$ $5\%$ of the computing resources.