Fractional step-method of characteristics for solving shallow-water type equations

Abstract

本文回顧如何以分布特性線法解類淺水波方程式，我們考慮淺水波在含摩擦力之任意形狀底床上之運動，分布法能夠把二維以上多變數耦合的類淺水波偏微分方程式，轉為數條一維的偏微分或常微分方程式，在解題的過程中，對流階段以特性線法處理，而非對流階段則以龍閣庫塔法或解析解法處理。其中特性線法的部分，我們把數條一維的耦合偏微分方程式轉為黎曼不變量的非耦合常微分方程式，而特性線與格點交錯的部分，則以內差法處理。本方法在一些類淺水波方程式的可應用性，則以數值結果與其他文獻上的實例來強化。This article gives a review of the formulation of fractional step-method of characteristics (FSMoC) for solving shallow-water type equations. The formulation is demonstrated by a shallow-water type equation with sources of force, arbitrary bottom topography and friction force. The fractional step-method (FS) splits the multidimensional shallow-water type equations into sequential augmented one-dimensional problems (PDEs or ODEs). Among the sequential problems, the advection phases are solved by the method of characteristics (MoC) and the non-advection phases are solved either by analytical methods or the Runge-Kutta method. In MoC, the one-dimensional PDEs are transformed to ordinary differential equations using Riemann invariants, which should be interpolated at each time step since the characteristic curves do not fall on a grid system. The applicability of the prescribed formulation is demonstrated by numerical results or strengthened by the studies in literature.