Higher-order accurate space-time schemes for computational astrophysics—Part I: finite volume methods
Higher-order accurate space-time schemes for computational astrophysics—Part I: finite volume methods
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DOI：
10.1007/s41115-017-0002-8
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发表时间：
2017-03
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影响因子：
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通讯作者：
D. Balsara
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文献类型：
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作者：
D. Balsara
随着计算天体物理学面临成为精确科学的压力，越来越需要转向计算天体物理学的高精度方案。计算天体物理学的算法需求确实非常特殊。这些方法需要是稳健的，并保持密度和压力的正性。相对论流动应该保持在亚光速。这些要求给计算天体物理学代码带来了额外的压力，而传统的流体动力学代码通常不会感受到这些压力。因此需要进行专门审查。重点介绍了加权基本无振荡（韦诺）格式、间断Galerkin（DG）格式和PNPM格式。韦诺格式是传统二阶有限体积格式的高阶推广。在三阶时，它们最类似于分段抛物型方法格式，这也包括在内。DG格式演化了解的所有矩，结果比韦诺格式更精确。PNPM方案介于韦诺和DG方案之间。它们演化出N阶空间多项式，同时重建高达M阶的高阶项。因此，时间步长可以更大。与时间相关的天体物理学代码需要在空间和时间上准确，因此空间和时间精度必须匹配。这是实现强稳定性保持龙格-库塔格式和ADER（任意导数在空间和时间）计划的帮助下，这两个也被描述。这篇评论的重点是计算机可实现的想法，不一定是基础理论。
As computational astrophysics comes under pressure to become a precision science, there is an increasing need to move to high accuracy schemes for computational astrophysics. The algorithmic needs of computational astrophysics are indeed very special. The methods need to be robust and preserve the positivity of density and pressure. Relativistic flows should remain sub-luminal. These requirements place additional pressures on a computational astrophysics code, which are usually not felt by a traditional fluid dynamics code. Hence the need for a specialized review. The focus here is on weighted essentially non-oscillatory (WENO) schemes, discontinuous Galerkin (DG) schemes and PNPM schemes. WENO schemes are higher order extensions of traditional second order finite volume schemes. At third order, they are most similar to piecewise parabolic method schemes, which are also included. DG schemes evolve all the moments of the solution, with the result that they are more accurate than WENO schemes. PNPM schemes occupy a compromise position between WENO and DG schemes. They evolve an Nth order spatial polynomial, while reconstructing higher order terms up to Mth order. As a result, the timestep can be larger. Time-dependent astrophysical codes need to be accurate in space and time with the result that the spatial and temporal accuracies must be matched. This is realized with the help of strong stability preserving Runge–Kutta schemes and ADER (Arbitrary DERivative in space and time) schemes, both of which are also described. The emphasis of this review is on computer-implementable ideas, not necessarily on the underlying theory.