Stability analysis and error estimates of local discontinuous Galerkin methods with semi-implicit spectral deferred correction time-marching for the Allen–Cahn equation

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Abstract

This paper is concerned with the stability and error estimates of the local discontinuous Galerkin (LDG) method coupled with semi-implicit spectral deferred correction (SDC) time-marching up to third order accuracy for the Allen–Cahn equation. Since the SDC method is based on the first order convex splitting scheme, the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step, which increases the difficulty of the theoretical analysis. For the LDG discretizations coupled with the second and third order SDC methods, we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimensional spaces. At the same time, the iteration and integral involved in the semi-implicit SDC scheme also increase the difficulty of the theoretical analysis. Comparing to the Runge–Kutta type semi-implicit schemes which exclude the left-most endpoint, the SDC scheme in this paper includes the left-most endpoint as a quadrature node. This makes the test functions of the SDC scheme more complicated and the energy equations are more difficult to construct. We provide two different ideas to overcome the difficulty of the nonlinear terms. By choosing the test functions carefully, the energy stability and error estimates are obtained in the sense that the time step Δt only requires a positive upper bound and is independent of the mesh size h. Numerical examples are presented to illustrate our theoretical results.

Introduction

Let Ω be a bounded domain with dimension d3 and 0<T<. We analyze the LDG discretizations coupled with semi-implicit SDC methods to the Allen–Cahn equation utΔu+1ε2f(u)=0,inΩ×(0,T],u(x,0)=u0(x),inΩwith the Neumann boundary condition uν=0,Ω×(0,T],where f(u)=Ψ(u) and Ψ(u)=14(1u2)2. Since it is well-known that the Allen–Cahn equation satisfies the maximum principle, as in [1], we assume maxuR|f(u)|C,where C is a positive constant. Here we might as well require C>1.

In order to describe the motion of anti-phase boundaries in crystalline solids, Allen and Cahn [2] originally propose the well-known Allen–Cahn equation. Consequently, numerous numerical efforts have been devoted to study the Allen–Cahn equation, for instance, finite difference methods [3], [4], finite element methods [5], [6], [7], [8], DG methods [9], [10] and LDG methods [11]. The first order time marching methods [3], [6], [8], [9], the first and second-order implicit-explicit (IMEX) methods [7], the implicit additive Runge–Kutta (ARK) and the diagonally implicit Runge–Kutta (DIRK) methods [10] are used for time discretization among the above papers. Guo et al. [11] present the LDG schemes for the Allen–Cahn equation coupled with semi-implicit SDC time discrete methods in [12].

For some partial differential equations (PDEs), especially those with nonlinear terms, we often need to take higher order numerical discretizations in space and time to get more accurate numerical solutions. Recently, the one-dimensional linear advection–diffusion equation [13], the one-dimensional nonlinear convection–diffusion equation [14] and the multi-dimensional nonlinear convection–diffusion equation [15] are considered to analyze the stability and error estimates for the LDG schemes with IMEX RK time discretizations up to third order accuracy, the authors obtain the stability and error results in the sense that the time step Δt is only required to be upper-bounded by a positive constant independent of the mesh size h. Song and Shu in [16] study the Cahn–Hilliard equation in one and two dimensions, and they prove the unconditional energy stability for a second-order IMEX LDG method. In Ref. [7], the authors consider the first and second-order IMEX finite element methods in one and multi-dimension and prove the corresponding energy stability in the similar sense as in [13], [14], [15]. The purpose of this paper is to study the stability and error estimates of the LDG schemes with second and third order SDC time discrete methods for the Allen–Cahn equation in one and multi-dimension.

The SDC method, as well as the integral deferred correction (InDC) method [17], is based on low order time integration methods and corrected iteratively. The SDC method is easy to construct for any order of accuracy in comparison with RK methods which are more difficult for constructing high order of accuracy. More general information about the semi-implicit SDC method coupled with LDG discretization can be found in [18], [19], and applications of the SDC method are presented in [1], [11], [20], [21]. Comparing to the InDC method [17], which uses uniform quadrature nodes excluding the left-most endpoint to guarantee high order accuracy increase and can be reformulated as an IMEX RK type scheme with the Butcher tableau, the SDC method allows Gauss nodes, Legendre–Gauss–Radau nodes, Legendre–Gauss–Lobatto nodes or Chebyshev nodes. In the SDC method of this paper, the left-most endpoint is included and used as a quadrature node. The test functions for stability analysis are more intricate when compared with those in the IMEX RK schemes.

As we know, the DG method is a kind of finite element method with discontinuous, piecewise polynomials as basis functions, which was first proposed by Reed and Hill in [22]. It has many advantages over other finite element methods, such as the usefulness of the highly nonuniform and unstructured meshes, the simple choice of the trial and test spaces, the dimension-independence in construction, and so on. For more information, we refer readers to see papers [23], [24], [25], [26]. By extending the DG method, Cockburn and Shu in [27] first put forward the LDG method for handling with PDEs which contain second order spatial derivative. The idea of the LDG method is to apply the DG method after rewriting the higher order equation into a system of first order equations. We refer to the linear cases [13], [28], [29], [30] and the nonlinear cases [31], [32], [33], [34], [35], [36] for a general information about the LDG method.

The main contribution of this paper is to prove the stability and error estimates of the LDG schemes coupled with the second and third order SDC time discretizations for the Allen–Cahn equation (1.1)–(1.2). Since the implicit treatment of the nonlinear item u3 results in a nonlinear system, we prove the unique solvability of the fully-discrete numerical discretizations by using the standard fixed point argument in finite dimensional spaces. Comparing to the Runge–Kutta type semi-implicit schemes [13] which exclude the left-most endpoint, the SDC scheme in this paper includes the left-most endpoint as a quadrature node. This makes the test functions of the SDC scheme more complicated and the energy equations are more difficult to construct. For implicit time-discrete schemes, there are always some positive items when analyzing the stability and error estimates. We make full use of this advantage to deal with the nonlinear items for the stability analysis of the second order SDC–LDG scheme. For the stability analysis of the third order SDC–LDG scheme and the error estimates of both SDC–LDG schemes, we use the property (1.3) to handle with the nonlinear items. By a careful selection of the test functions, the energy stability and error estimates for the second and third order time-discrete LDG schemes are obtained in the sense that the time step Δt requires only a positive upper bound and is independent of the mesh size h.

The rest of the paper is organized as follows. In Section 2, we introduce some notations, projections and the SDC scheme that will be used in the following analysis. In Section 3, we present the LDG scheme with second order semi-implicit SDC method for the Allen–Cahn equation (1.1). The unique solvability, stability and error estimates for the second order SDC–LDG discretization are proved. Similar analysis for the third order SDC–LDG discretization is presented in Section 4. In Section 5, some numerical results are provided to verify the theoretical analysis. Concluding remarks are given in Section 6.

Section snippets

Preliminaries

In this section, we introduce the finite element spaces, some notations and definition norms, and the SDC scheme to be used later in the paper. We also present some projections and certain corresponding interpolation properties for the finite element spaces which will be used for the error analysis.

LDG method with the second order SDC scheme

In this section, we present the second order time accurate SDC–LDG scheme for the Allen–Cahn equation (1.1)–(1.2) in ΩRd with d3.

The LDG method with the third order SDC scheme

In this section, we aim at discussing the third order time accurate SDC–LDG scheme for the Allen–Cahn equation (1.1)–(1.2) in ΩRd with d3.

Numerical results

In this section, we provide some numerical results to confirm our theoretical analysis. For more computational simulation, we refer readers to see paper [11].

Conclusion

The semi-implicit SDC method is effective and robust when solving nonlinear PDEs. In addition, it is easy to construct for any order of accuracy, with the order increased for each iteration. And the LDG method is easy to handle with complicated geometry and has the attractive property of parallel efficiency. Motivated by the above properties, by discretizing space with LDG method and discretizing time with SDC method, we construct the fully-discrete numerical schemes with second and third time

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    Research supported by Science Challenge Project TZZT2019-A2.3, National Numerical Windtunnel grants NNW2019ZT4-B08, NSFC grants 11722112.

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