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首页> 外文期刊>Journal of Computational Physics >Upwind schemes for the wave equation in second-order form
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Upwind schemes for the wave equation in second-order form

机译：二阶形式的波动方程的迎风格式

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We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d'Alembert's exact solution. We construct conservative finite difference approximations, although finite volume approximations are also possible. High-order accuracy is obtained using a space-time procedure which requires only two discrete time levels. The advantages of our approach include efficiency in both memory and speed together with accuracy and robustness. The stability and accuracy of the approximations in one and two space dimensions are studied through normal-mode analysis. The form of the dissipation and dispersion introduced by the schemes is elucidated from the modified equations. Upwind schemes are implemented and verified in one dimension for approximations up to sixth-order accuracy, and in two dimensions for approximations up to fourth-order accuracy. Numerical computations demonstrate the attractive properties of the approach for solutions with varying degrees of smoothness.
机译：我们为二阶形式的波动方程开发了新的高阶精确迎风方案。这些方案是针对二阶形式的方程式直接开发的，与将方程式转换为一阶双曲系统相反。这些方案基于使用d'Alembert精确解的局部Riemann型问题的解。尽管有限体积近似也是可行的，但我们构造了保守的有限差分近似。使用仅需两个离散时间级别的时空过程即可获得高阶精度。我们方法的优点包括存储效率和速度方面的效率以及准确性和鲁棒性。通过正模分析研究了一维和二维空间中近似值的稳定性和准确性。从修改的方程式中阐明了方案引入的耗散和色散形式。在一维上实现和验证迎风方案，以达到六阶精度的近似值，在二维中实现并验证对风方案，以近似四阶精度的形式实现。数值计算证明了该方法对于具有不同平滑度的解决方案的吸引人的特性。

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