High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid
High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid
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DOI：
10.1016/j.jcp.2021.110792
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发表时间：
2021-01
期刊：
影响因子：
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通讯作者：
S. Gaudreault;M. Charron;V. Dallerit;M. Tokman
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文献类型：
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作者：
S. Gaudreault;M. Charron;V. Dallerit;M. Tokman
本文提出了一种新的数值方法，利用空间和时间的高阶数值离散来求解球上的浅水方程。时空张量形式主义是用来表示协变的运动方程，并描述旋转立方体球网格的几何形状。空间离散化与直接通量重建方法，这是一种替代配方的间断伽辽金方法。运动方程以微分形式求解，所得离散化不受求积规则的约束。众所周知，传统显式方法的时间步长受到最快波相速度的限制。采用指数积分，使积分具有显着更大的时间步长，并提高整体时间积分的效率。构造了4、5、6阶多步指数传播迭代法，并应用于浅水波方程的时间积分。这些新的计划，使时间积分高阶精度，但没有显着增加计算时间相比，低阶方法。在基于Krylov的KIOPS（Krylov with imcomplete orthogonalization procedure solver）算法中，指数格式中的指数矩阵函数-向量乘积采用Jacobian的复步长近似。新的数值方法的性能进行评估，使用一组标准的基准测试。
A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion covariantly and to describe the geometry of the rotated cubed-sphere grid. The spatial discretization is done with the direct flux reconstruction method, which is an alternative formulation to the discontinuous Galerkin approach. The equations of motion are solved in differential form and the resulting discretization is free from quadrature rules. It is well known that the time step of traditional explicit methods is limited by the phase velocity of the fastest waves. Exponential integration is employed to enable integrations with significantly larger time step sizes and improve the efficiency of the overall time integration. New multistep-type exponential propagation iterative methods of orders 4, 5 and 6 are constructed and applied to integrate the shallow-water equations in time. These new schemes enable time integration with high-order accuracy but without significant increases in computational time compared to low-order methods. The exponential matrix functions-vector products used in the exponential schemes are approximated using the complex-step approximation of the Jacobian in the Krylov-based KIOPS (Krylov with incomplete orthogonalization procedure solver) algorithm. Performance of the new numerical methods is evaluated using a set of standard benchmark tests.