Elsevier

Journal of Computational Physics

Volume 338, 1 June 2017, Pages 269-284
Journal of Computational Physics

Semi-implicit spectral deferred correction methods for highly nonlinear partial differential equations

https://doi.org/10.1016/j.jcp.2017.02.059Get rights and content

Abstract

The goal of this paper is to develop a novel semi-implicit spectral deferred correction (SDC) time marching method. The method can be used in a large class of problems, especially for highly nonlinear ordinary differential equations (ODEs) without easily separating of stiff and non-stiff components, which is more general and efficient comparing with traditional semi-implicit SDC methods. The proposed semi-implicit SDC method is based on low order time integration methods and corrected iteratively. The order of accuracy is increased for each additional iteration. And we also explore its local truncation error analytically. This SDC method is intended to be combined with the method of lines, which provides a flexible framework to develop high order semi-implicit time marching methods for nonlinear partial differential equations (PDEs). In this paper we mainly focus on the applications of the nonlinear PDEs with higher order spatial derivatives, e.g. convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the fully discrete schemes are all high order accurate in both space and time, and stable numerically with the time step proportional to the spatial mesh size. Numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi-implicit SDC method.

Introduction

In this paper, we will develop a novel semi-implicit spectral deferred correction (SDC) time marching method for solving time dependent highly nonlinear partial differential equations (PDEs) containing high order spatial derivatives. Coupling with the local discontinuous Galerkin (LDG) spatial discretization, we develop an arbitrary high order accurate and stable schemes for convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Due to the local property of the LDG methods, the resulting implicit scheme is easy to implement and can be solved in an explicit way when it is coupled with iterative methods.

The discontinuous Galerkin (DG) method is a class of finite element methods, in which using a completely discontinuous piecewise polynomials as the numerical solution and the test spaces. It was first designed as a method for solving hyperbolic conservation laws containing only first order spatial derivatives, e.g. Reed and Hill [26] for solving linear equations, and Cockburn et al. [5], [6], [7], [8] for solving nonlinear problems.

It is difficult to apply the DG method directly to PDEs containing higher order spatial derivatives, therefore the LDG method was introduced. The idea of the LDG method is to rewrite the equations with higher order derivatives as a first order system, then apply the DG method to the system. The first LDG method was constructed by Cockburn and Shu [9] for solving nonlinear convection diffusion equations containing second order spatial derivatives. Then LDG methods have been successfully designed and applied in a number of models involving diffusion and dispersive problems (see for example the review paper [33]). DG and LDG methods also have several attractive properties, for example: allowing for efficient h, p adaptivity and having excellent parallel efficiency. The most important property of DG and LDG methods is high order accurate, which motivates us to develop high order temporal accuracy scheme to get the goal of obtaining high order accuracy in both space and time together with robust stability conditions.

By the method of lines, the application of the LDG method for spatial variables for a partial differential equation will generate a large coupled system of ordinary differential equations (ODEs). The development of suitable solvers for solving different types of ODEs have attracted a lot of attention in the last decades. For non-stiff ODEs, there exist extremely effective high order explicit methods [1], [4], [21], such as Runge–Kutta, linear multi-step and predictor corrector methods. For stiff problems, the situation is considerably more complicated, but still many efficient implicit methods [1], [4], [22] have been developed. Nevertheless, Dutt et al. constructed a new variation of the classical method of deferred corrections, the SDC method in [13], which preserve good stability and accuracy properties for stiff problems. Tang et al. provided a general framework for the convergence of the SDC method in [27].

In some cases, the resulting ODEs include both stiff term FS and non-stiff term FN, and can be written as{ut=FS(t,u(t))+FN(t,u(t)),t[0,T]u(0)=u0. An efficient time marching technique to solve the ODEs (1.1) is semi-implicit methods, which treats the stiff component FS implicitly and the non-stiff component FN explicitly. For example, Minion developed an efficient semi-implicit SDC method for solving the ODEs (1.1). Xia, Xu and Shu [29] explored the SDC method, the additive Runge–Kutta (ARK) method and the exponential time differencing (ETD) method for the LDG methods to solve PDEs with higher order spatial derivatives, which were all validated to be effective. The existing semi-implicit SDC methods have been developed to solve many problems, such as the phase field problems [15], [24], [31] and the phase field crystal equation [19]. Unfortunately, these semi-implicit time marching methods are mainly efficient for problems with easily separate stiff and non-stiff components.

However, it is not always easy to separate the stiff and non-stiff components, and therefore the use of traditional semi-implicit schemes is not straightforward. In such cases, one usually relies on fully implicit schemes. But, fully implicit schemes have the disadvantages of difficult implementation and poor stability properties. To get a suitable solver for this class of problems, Boscarino et al. [2] developed several semi-implicit Runge–Kutta schemes up to order three for time-dependent highly nonlinear PDEs. Guo, Filbet and Xu [20] applied the semi-implicit Runge–Kutta methods coupled with LDG spatial discretization to solve a series of highly nonlinear PDEs containing higher order spatial derivatives, obtaining second order and third order accuracy in both time and space. Motivated by this idea and the desire to develop higher order temporal accuracy, we will propose a novel semi-implicit SDC method for highly nonlinear PDEs, namely, the stiff and non-stiff components can not be well separated. The semi-implicit Runge–Kutta method introduced in [20] and the SDC method proposed here are all efficient for highly nonlinear PDEs. However, the Runge–Kutta method has some limitations, for example, it is more difficult to construct for higher order accuracy. While for the SDC method, an advantage of this method is that it is a one step method and can be constructed easily and systematically for any order of accuracy.

In this paper, we will apply the proposed semi-implicit SDC method to solve a series of highly nonlinear time dependent PDEs, namely, the nonlinear convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation with degenerate mobility, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Convex splitting schemes were developed by Guo and Xu for the Cahn–Hilliard equation [17], the Cahn–Hilliard–Brinkman system [18] and the phase field crystal equation [19], which were unconditionally energy stable. However, these schemes are only first order or second order accurate in time. Based on the first order schemes, the proposed semi-implicit SDC method can therefore be applied to improve the temporal accuracy. The surface diffusion and Willmore flow of graphs are both highly nonlinear fourth-order PDEs, high order semi-implicit time marching methods are difficult to construct and therefore are desirable to develop.

The organization of the paper is as follows. In Section 2, we develop a novel high order semi-implicit SDC method for a general class of ODEs and study its local truncation error analytically. Numerical experiments are carried out in Section 3, testing the performance of the semi-implicit SDC method coupled with the LDG spatial discretization for solving a series of highly nonlinear PDEs, including the nonlinear convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation with degenerate mobility, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Finally, we give concluding remarks in Section 4.

Section snippets

The semi-implicit SDC methods

The SDC method was first constructed by Dutt, Greengard and Rokhlin [13] to develop high order stable methods for stiff and non-stiff problems. Then, a semi-implicit SDC method was introduced in [25] to solve ODEs containing both stiff and non-stiff components. The method is very efficient for PDEs with easily separate stiff and non-stiff components, which treats the stiff terms implicitly and the non-stiff terms explicitly. The classical semi-implicit SDC method proposed in [25] will first be

Applications

In this section, we perform numerical experiments for a series of highly nonlinear PDEs, including the nonlinear convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation with degenerate mobility, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. These examples are used to verify that our spatial and time discretization methods can achieve high order accuracy in both space and time for PDEs without easily separating stiff

Concluding remarks

We presented in this paper a novel semi-implicit spectral deferred correction (SDC) time marching method, which are motivated by the desire to design higher order temporal methods for PDEs without easily separating of stiff and non-stiff components. Combined with the local discontinuous Galerkin method, this SDC method has been applied on a series of highly nonlinear PDEs, including the nonlinear convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard

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    Research supported by NSFC grant No. 11601490.

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    Research supported by NSFC grant Nos. 11471306, 11371342.

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    Research supported by NSFC grant Nos. 11371342, 11626253, 91630207.

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