On Energy Stable Runge-Kutta Methods for the Water Wave Equation and its Simplified Non-Local Hyperbolic Model  

作　　者：Lei Li Jian-Guo Liu Zibu Liu Yi Yang Zhennan Zhou 

机构地区：[1]School of Mathematical Sciences,Institute of Natural Sciences,MOE-LSC,Shanghai Jiao Tong University,Shanghai,200240,P.R.China [2]Department of Mathematics and Department of Physics,Duke University,Durham,NC 27708,USA [3]Department of Mathematics,Duke University,Durham,NC 27708,USA [4]Nanjing Research Institute of Electronics Technology,Nanjing,P.R.China [5]Beijing International Center for Mathematical Research,Peking University,P.R.China

出　　处：《Communications in Computational Physics》2022年第6期222-258,共37页计算物理通讯（英文）

基　　金：L.Li was supported by National Natural Science Foundation of China(Grant No.31571071);Shanghai Sailing Program 19YF1421300;J.-G.Liu was supported in part by DMS-2106988;Z.Zhou was supported by the National Key R&D Program of China,Project Number 2021YFA001200;the NSFC,grant number 12171013.

摘　　要：Although interest in numerical approximations of the water wave equationgrows in recent years, the lack of rigorous analysis of its time discretization inhibits thedesign of more efficient algorithms. In practice of water wave simulations, the tradeoff between efficiency and stability has been a challenging problem. Thus to shed lighton the stability condition for simulations of water waves, we focus on a model simpli-fied from the water wave equation of infinite depth. This model preserves two mainproperties of the water wave equation: non-locality and hyperbolicity. For the constantcoefficient case, we conduct systematic stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and generalRunge-Kutta methods in time. As a result, an optimal time discretization strategy isprovided in the form of a modified CFL condition, i.e. ∆t = O(√∆x). Meanwhile,the energy stable property is established for certain explicit Runge-Kutta methods.This CFL condition solves the problem of efficiency and stability: it allows numericalschemes to stay stable while resolves oscillations at the lowest requirement, which onlyproduces acceptable computational load. In the variable coefficient case, the convergence of the semi-discrete approximation of it is presented, which naturally connectsto the water wave equation. Analogue of these results for the water wave equationof finite depth is also discussed. To validate these theoretic observation, extensive numerical tests have been performed to verify the stability conditions. Simulations of thesimplified hyperbolic model in the high frequency regime and the water wave equation are also provided.

关 键 词：Runge-Kutta methods NON-LOCALITY HYPERBOLICITY 

分 类 号：O17[理学—数学]