Well-Balanced High-Order Centred Schemes for Non-Conservative Hyperbolic Systems. Applications to Shallow Water Equations with Fixed and Mobile Bed
Well-Balanced High-Order Centred Schemes for Non-Conservative Hyperbolic Systems. Applications to Shallow Water Equations with Fixed and Mobile Bed
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DOI：
10.1016/j.advwatres.2009.02.006
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发表时间：
2009-06
影响因子：
4.7
通讯作者：
A. Canestrelli;A. Siviglia;M. Dumbser;E. Toro
中科院分区：
文献类型：
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作者：
A. Canestrelli;A. Siviglia;M. Dumbser;E. Toro
本文发展了一种高阶精度的中心格式，用于数值求解含有非保守乘积和源项的一维双曲型方程组。结合在[Toro E，Siviglia A. PRICE：双曲方程组的原始中心格式。Int J Numer Methods Fluids 2003;42：1263-91]与最近开发的路径保守方案[Castro M，Gallardo J，Parés C.基于状态重构的高阶有限体积格式求解具有非保守积的双曲方程组在浅水系统中的应用。Math Comput 2006;75：1103-34; Parés C.非保守双曲方程组的数值方法：理论框架。SIAM J Numer Anal 2006;44：300-21]，我们提出了一种新的PRICE-C格式，如果潜在的PDE系统是守恒律，则该格式会自动简化为修改的守恒FORCE格式。然后，通过ADER方法和韦诺重建技术，将由此产生的一阶精确中心方法扩展到空间和时间的高阶精确度。研究了浅水方程的PRICE-C方法的平衡性。最后，将新格式应用于具有固定底地形的浅水方程组和具有可变底地形的浅水方程组，并求解了附加的泥沙输运方程。
This paper concerns the development of high-order accurate centred schemes for the numerical solution of one-dimensional hyperbolic systems containing non-conservative products and source terms. Combining the PRICE-T method developed in [Toro E, Siviglia A. PRICE: primitive centred schemes for hyperbolic system of equations. Int J Numer Methods Fluids 2003;42:1263–91] with the theoretical insights gained by the recently developed path-conservative schemes [Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products applications to shallow-water systems. Math Comput 2006;75:1103–34; Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21], we propose the new PRICE-C scheme that automatically reduces to a modified conservative FORCE scheme if the underlying PDE system is a conservation law. The resulting first-order accurate centred method is then extended to high order of accuracy in space and time via the ADER approach together with a WENO reconstruction technique. The well-balanced properties of the PRICE-C method are investigated for the shallow water equations. Finally, we apply the new scheme to the shallow water equations with fix bottom topography and with variable bottom solving an additional sediment transport equation.