High-Order CESE Methods for the Euler Equations
High-Order CESE Methods for the Euler Equations
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DOI：
10.2514/6.2011-298
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发表时间：
2011-01
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影响因子：
--
通讯作者：
D. Bilyeu
中科院分区：
文献类型：
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作者：
D. Bilyeu
最近，Chang 1报道了一类求解非线性双曲型偏微分方程的新的高阶CESE方法。一系列的高阶算法已经开发的基础上，系统的，递归的制定，达到四阶，六阶和八阶精度。新的高阶CESE方法共享原始二阶CESE方法的许多有利属性，包括：（i）仅涉及其中寻求解的节点周围的直接网格节点的紧凑网格模板，（ii）CFL稳定性约束保持相同，即，≤ 1，与原来的二阶方法相比，和（iii）高超的冲击捕获能力，而无需使用近似黎曼解。通过求解Burger方程证明了新算法的有效性。本文推广了Chang的高阶方法，使之适用于线性和非线性双曲型偏微分方程组。给出了求解耦合方程组的一般公式，具有任意高阶精度。为了证明配方，几个线性和非线性的情况下报告。首先，我们求解一个带源项的对流方程和线性声学方程。然后，我们解决声波，爆炸波，和舒和Osher的声波与冲击波相互作用的测试情况下的欧拉方程。数值结果表明，高阶收敛的连续网格加密。
Recently, Chang 1 reported a new class of high-order CESE methods for solving nonlinear hyperbolic partial differential equations. A series of high-order algorithms have been developed based on a systematic, recursive formulation that achieves fourth-, sixth-, and eighth-order accuracy. The new high-order CESE method shares many favorable attributes of the original second- order CESE method, including: (i) compact mesh stencil involving only the immediate mesh nodes surrounding the node where the solution is sought, (ii) the CFL stability constraint remains to be the same, i.e., ≤ 1, as compared to the original second-order method, and (iii) superb shock capturing capability without using an approximate Riemann solver. The new algorithm has been demonstrated by solving Burger’s equation. In the present paper, we extend Chang’s high-order method for system of linear and nonlinear hyperbolic partial differential equations. A general formulation is presented for solving the coupled equations with arbitrarily high-order accuracy. To demonstrate the formulation, several linear and nonlinear cases are reported. First, we solve a convection equation with source term and the linear acoustics equations. We then solve the Euler equations for acoustic waves, a blast wave, and Shu and Osher’s test case for acoustic waves interacting with a shock. Numerical results show higher-order convergence by continuous mesh refinement.