Fast operator splitting methods for nonlinear PDEs.

摘要

Operator splitting methods have been applied to nonlinear partial differential equations that involve operators of different nature. The main idea of these methods is to decompose a complex equation into simpler sub-equations, which can be solved separately. The main advantage of the operator splitting methods is that they provide a great flexibility in choosing different numerical methods, depending on the feature of each sub-problem. In this dissertation, we have developed highly accurate and efficient numerical methods for several nonlinear partial differential equations, which involve both linear and nonlinear operators.;We first propose a fast explicit operator splitting method for the modified Buckley-Leverett equations which include a third-order mixed derivatives term resulting from the dynamic effects in the pressure difference between the two phases. The method splits the original equation into two equations, one with a nonlinear convective term and the other one with high-order linear terms so that appropriate numerical methods can be applied to each of the split equations: The high-order linear equation is numerically solved using a pseudo-spectral method, while the nonlinear convective equation is integrated using the Godunov-type central-upwind scheme. The spatial order of the central-upwind scheme depends on the order of the piecewise polynomial reconstruction: We test both the second-order minmod-based reconstruction and fifth-order WENO5 one to demonstrate that using higher-order spatial reconstruction leads to more accurate approximation of solutions.;We then propose fast and stable explicit operator splitting methods for two phase-field models (the molecular beam epitaxy equation with slope selection and the Cahn-Hilliard equation), numerical simulations of which require long time computations. The equations are split into nonlinear and linear parts. The nonlinear part is solved using a method of lines combined with an efficient large stability domain explicit ODE solver. The linear part is solved by a pseudo-spectral method, which is based on the exact solution and thus has no stability restriction on the time step size.;We have verified the numerical accuracy of the proposed methods and demonstrated their performance on extensive one- and two-dimensional numerical examples, where different solution profiles can be clearly observed and are consistent with previous analytical studies.