Abstract For Euler equations defined in a domain Ω with boundary conditions specified on a curved boundary Γ , we consider a discontinuous Galerkin (DG) method with high order polynomial basis functions on a geometry fitting triangular and tetrahedral grids. We derive a modification to the DG scheme defined on a boundary triangle and tetrahedron based on the general curvilinear element DG method. In the modified DG scheme, even though integration along the curved boundary is still necessary, integrals over any curvilinear element are avoided. To further simplify the calculation of integration along the curved boundary, the modified DG scheme is constructed by the relationship between normal vector of geometric boundary and surface Jacobian. In the new modified DG scheme, not only integrals over any curvilinear element are avoided, but also the line and face integrations along the curved boundary are not required. Numerical tests suggest that such a new modified DG scheme is as accurate as the basic modified DG scheme and full curvilinear DG scheme and more efficient than latter, and is stable on reasonably coarse, as well as more refined, grids. It is shown that this new modified DG scheme can be extremely useful to solve the problems defined in a domain with boundary conditions specified on a curved boundary.

中文翻译：

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应用于三角形和四面体网格上的欧拉方程的不连续伽辽金方法的曲线边界处理

摘要 对于定义在域 Ω 中且边界条件指定在弯曲边界 Γ 上的欧拉方程，我们考虑了在几何拟合三角和四面体网格上具有高阶多项式基函数的不连续伽辽金 (DG) 方法。我们基于一般曲线单元 DG 方法对在边界三角形和四面体上定义的 DG 方案进行了修改。在修改后的 DG 方案中，尽管沿曲线边界积分仍然是必要的，但避免了对任何曲线元素的积分。为进一步简化沿曲线边界的积分计算，通过几何边界法向量与表面雅可比矩阵的关系构造改进的DG方案。在新修改的 DG 方案中，不仅避免了对任何曲线元素的积分，但也不需要沿曲线边界的线和面积分。数值试验表明，这种新的修改 DG 方案与基本修改 DG 方案和全曲线 DG 方案一样准确，并且比后者更有效，并且在合理粗略和更精细的网格上稳定。结果表明，这种新的修改后的 DG 方案对于解决在曲线边界上指定边界条件的域中定义的问题非常有用。