A curved boundary treatment for discontinuous Galerkin schemes solving time dependent problems

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• 作者：Xiangxiong Zhang ^{zhan1966@purdue.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Discontinuous Galerkin ; Curved boundary ; Local discontinuous Galerkin method ; High order accuracy ; Conservation laws ; Wave equations
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：308
• 期：Complete
• 页码：153-170
• 全文大小：940 K

文摘

For problems defined in a two-dimensional domain Ω with boundary conditions specified on a curve Γ, we consider discontinuous Galerkin (DG) schemes with high order polynomial basis functions on a geometry fitting triangular mesh. It is well known that directly imposing the given boundary conditions on a piecewise segment approximation boundary class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999115008554&_mathId=si1.gif&_user=111111111&_pii=S0021999115008554&_rdoc=1&_issn=00219991&md5=f24d8dbf7510534c81913072d9f24d25" title="Click to view the MathML source">Γ_hclass="mathContainer hidden">class="mathCode">${\mathrm{\Gamma }}_h$ will render any finite element method to be at most second order accurate. Unless the boundary conditions can be accurately transferred from Γ to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999115008554&_mathId=si1.gif&_user=111111111&_pii=S0021999115008554&_rdoc=1&_issn=00219991&md5=f24d8dbf7510534c81913072d9f24d25" title="Click to view the MathML source">Γ_hclass="mathContainer hidden">class="mathCode">${\mathrm{\Gamma }}_h$, in general curvilinear element method should be used to obtain high order accuracy. We discuss a simple boundary treatment which can be implemented as a modified DG scheme defined on triangles adjacent to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999115008554&_mathId=si1.gif&_user=111111111&_pii=S0021999115008554&_rdoc=1&_issn=00219991&md5=f24d8dbf7510534c81913072d9f24d25" title="Click to view the MathML source">Γ_hclass="mathContainer hidden">class="mathCode">${\mathrm{\Gamma }}_h$. Even though integration along the curve is still necessary, integrals over any curved element are avoided. If the domain Ω is convex, or if Ω is nonconvex and the true solutions can be smoothly extended to the exterior of Ω, the modified DG scheme is high order accurate. In these cases, numerical tests on first order and second order partial differential equations including hyperbolic systems and the scalar wave equation suggest that it is as accurate as the full curvilinear DG scheme.