In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.
翻译:在本文中,我们开发并展示了一种任意高顺序、高度平衡的量的WENO方法,同时对浅水方程式采用了修改过的Patankar延迟校正(MPDEC)时间整合方法。由于MPDEC的活性-保护特性,由此产生的方案是无条件保护水高度。要应用 MPDEC 方法,我们必须从生产-销毁系统的角度解释空间半分化。在传统的WENO 实施过程中,只需要稍作修改,我们才能解释它是如何做到的。在以第五顺序方法为重点的数字模拟中,我们展示了新方法的良好性能,并核实了理论性能。