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首页> 外文期刊>Journal of Computational Physics >Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors
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Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors

机译：基于局部预测变量的不连续Galerkin和有限体积方案的显式一步时间离散化

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We consider a family of explicit one-step time discretizations for finite volume and discontinuous Galerkin schemes, which is based on a predictor-corrector formulation. The predictor remains local taking into account the time evolution of the data only within the grid cell. Based on a space-time Taylor expansion, this idea is already inherent in the MUSCL finite volume scheme to get second order accuracy in time and was generalized in the context of higher order ENO finite volume schemes. We interpret the space-time Taylor expansion used in this approach as a local predictor and conclude that other space-time approximate solutions of the local Cauchy problem in the grid cell may be applied. Three possibilities are considered in this paper: (1) the classical space-time Taylor expansion, in which time derivatives are obtained from known space-derivatives by the Cauchy-Kovalewsky procedure; (2) a local continuous extension Runge-Kutta scheme and (3) a local space-time Galerkin predictor with a version suitable for stiff source terms. The advantage of the predictor-corrector formulation is that the time evolution is done in one step which establishes optimal locality during the whole time step. This time discretization scheme can be used within all schemes which are based on a piecewise continuous approximation as finite volume schemes, discontinuous Galerkin schemes or the recently proposed reconstructed discontinuous Galerkin or PNPM schemes. The implementation of these approaches is described, advantages and disadvantages of different predictors are discussed and numerical results are shown.
机译：我们考虑了一系列基于有限量和不连续Galerkin方案的显式单步时间离散化方法，该方法基于预测器-校正器公式。考虑到仅在网格单元内数据的时间演变，预测变量将保持局部状态。基于时空泰勒展开，该思想已经在MUSCL有限体积方案中固有，可以及时获得二阶精度，并且在更高阶ENO有限体积方案中得到了推广。我们将在这种方法中使用的时空泰勒展开解释为局部预测变量，并得出结论，可以应用网格单元中局部柯西问题的其他时空近似解。本文考虑了三种可能性：（1）经典的时空泰勒展开式，其中时间导数是通过柯西-科瓦列斯基方法从已知的空间导数获得的; （2）局部连续扩展Runge-Kutta方案和（3）局部时空Galerkin预测器，其版本适用于刚性源项。预测器-校正器公式的优点在于，时间演化是一步完成的，这一步在整个时间步中建立了最佳位置。该时间离散化方案可以在基于分段连续逼近的所有方案中使用，如有限体积方案，不连续的Galerkin方案或最近提出的重建的不连续的Galerkin或PNPM方案。描述了这些方法的实现，讨论了不同预测变量的优缺点，并显示了数值结果。

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