Elsevier

Applied Numerical Mathematics

Volume 156, October 2020, Pages 242-264
Applied Numerical Mathematics

Convergence results for implicit–explicit general linear methods

https://doi.org/10.1016/j.apnum.2020.04.005Get rights and content

Abstract

This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.

Introduction

Consider the initial value problem for an autonomous system of differential equations in the formy(t)=f(y),t0ttF,y(t0)=y0, with y(t)Rm and f:RmRm Lipschitz continuous.

General linear methods (GLMs) [7] are numerical discretizations of (1) that generalize both Runge-Kutta methods (by computing multiple internal stages) and linear multistep methods (by transferring from one step to the next multiple pieces of information). One step of the GLM applied to (1) reads [6], [8], [9], [26]:Yi[n]=hj=1sai,jf(Yj[n])+j=1rui,jyj[n1],i=1,,s,yi[n]=hj=1sbi,jf(Yj[n])+j=1rvi,jyj[n1],i=1,,r, where h is the step size. The method (2a), (2b) can be written in vector form:Y[n]=h(AIm)f(Y[n])+(UIm)y[n1],y[n]=h(BIm)f(Y[n])+(VIm)y[n1], where Im is an identity matrix of the dimension of the ODE system, the abscissa vector is cRs, and the four coefficient matrices are ARs×s, URs×r, BRr×s and VRr×r. The method can be represented compactly in the following Butcher tableau:cAUBV. The method (3a), (3b) builds s internal stages Yi[n] which are meant to be order q approximations of the exact solution at the abscissae:Y[n]=y(tn1+ch)+O(hq+1), and r external stages y[n] which are constructed to be order p approximations of linear combinations of solution derivatives:y[n]=(WIm)ηp(h,y,tn)+O(hp+1), where the matrix W and the Nordsieck vector ηp are:W=[w0w1wp]Rr×(p+1),ηp(h,y,t)[y(t)Thp(y(p)(t))T]T.

Theorem 1 GLM classical order conditions [26]

A GLM (2a), (2b) is called preconsistent if [26]:Uw0=1s×1,Vw0=w0. A preconsistent GLM (2a), (2b) has order p (6) and stage order q=p1 or q=p (5) if and only if:c×kk!Ac×(k1)(k1)!Uwk=0,k=1,,q,=0kwk!Bc×(k1)(k1)!Vwk=0,k=1,,p, where c×k denotes the component-wise power operation.

Application of the GLM (2a), (2b) to the Dahlquist model problem for error propagation y=λy leads a numerical solution of the formy[n]=M(z)y[n1],M(z)=V+zB(I+zA)1U, where z=hλ and M(z)Rr×r is the stability matrix of the method. The stability region of the method is defined as:S={zC:supnM(z)nC}.

Many multiphysics systems of interest in science and engineering (1) are driven by stiff (fast) and non-stiff (slow) processes acting simultaneously. The implicit-explicit (IMEX) approach seeks to obtain efficient numerical solutions by solving the stiff components with an implicit, numerically stable scheme, and the non-stiff component with an explicit, computationally efficient scheme. The two schemes need to be coordinated carefully in order to obtain the desired accuracy and stability properties of the overall discretization.

IMEX schemes have been developed in the context of linear multistep methods [2], [19], [24], [32], Runge-Kutta methods [1], [3], [17], [18], [28], [29], [31], [40], [48], linearly-implicit Rosenbrock methods [12], [48], and extrapolation methods [16].

The Partitioned GLM (PGLM) framework was developed by Sandu and collaborators over a series of papers [42], [43], [45], [46], [47]. Using this framework they constructed the first IMEX GLM schemes, and provided coefficients for methods of order two [42] to six [44], [47].

Building on this work, Cardone at el. [13] proposed an extrapolation approach to construct IMEX GLMs: start with the implicit method, and replace the non-stiff function values at each stage by extrapolated values based on the previous step. High order, stable IMEX GLMs have been constructed by Jackiewicz and co-workers [14], [25], [27]. Bras et al. [4] have constructed IMEX GLMs with inherited Runge Kutta stability property. Bras et al. [5] have studied the local truncation error for IMEX GLMs of order q=p up to, and including, terms of order O(hp+2). Lang and Hundsdorfer used the extrapolation approach to construct IMEX GLMs of PEER type [30]. Schneider et al. then extended the approach to superconvergent IMEX PEER schemes [33], [34]. Soleimani et al. also constructed new IMEX PEER methods of high order [37], [38].

This paper performs a convergence study of IMEX general linear methods for step sizes that are small enough, but do not depend on the stiffness of the problem. To the best of author's knowledge, such results were not previously available in the literature.

  • 1.

    We first carry out a classical convergence analysis and conclude that IMEX GLMs of high stage order converge with the theoretical order for sufficiently small step sizes; the novelty of our analysis is that the step upper bounds depend on the method coefficients and on non-stiff Lipschitz constants, but are independent of the stiffness level of the fast component.

  • 2.

    Next, we carry out a convergence analysis for IMEX GLMs applied to singular perturbation problems. This analysis follows the approach of Schneider [35] for implicit GLMs. We show that the method has a unique numerical solution. We study the convergence for index-1 DAE problems, and then the convergence for very stiff singular perturbation problems. Under mild assumptions high stage order IMEX GLMs converge with the full theoretical order for both the non-stiff and the algebraic/stiff variables.

The remainder of the paper is organized as follows. A review of implicit-explicit general linear methods in the partitioned general linear method framework is provided in Section 2. Section 3 provides a classical convergence analysis for IMEX GLMs. Convergence results for IMEX GLMs applied to a singular perturbation problem are given in Section 7. Section 8 discusses the findings of the paper.

Section snippets

Implicit-explicit general linear methods for component and additively partitioned systems of differential equations

Here we focus on partitioned GLMs [42], [43], [44], [46], [47] applied to integrate systems with two components: a non-stiff one described by a right hand side function

, and a stiff one described by a right hand side function
. We consider partitioned GLMs where an explicit method used to solve the non-stiff component is paired with an implicit method to solve the stiff component. In this section we follow the presentation in [43], [44].

Convergence analysis for additive IMEX GLMs

We next study convergence for a fixed-step h solution. Consider a partitioned system (15) where the nonstiff component is Lipschitz-continuous with a moderate Lipschitz constant in a vicinity of the exact solution: The Lipschitz constant of the stiff component can be arbitrarily large. To avoid using it we make the following assumption.

Assumption 1 Separability of stiffness

Assume that, for any τ[t0,tF], there is an interval [τε,τ+ε] such that the implicit component can be locally decomposed into a linear part and a nonlinear

The index-1 differential algebraic problem

Consider the index-1 differential algebraic equation (DAE) [20], [22], [39]:{x=f(x,z),0=g(x,z), where f, g are smooth functions and the sub-Jacobian gz is invertible in a neighborhood of the solution. The initial values [x0,z0] are consistent if g(x0,z0)=0. By the implicit function theorem the algebraic equation can be locally solved uniquely to express z as a function of x:z=G(x). Replacing this in the differential equation (20) leads to the following reduced ODE:

The singular perturbation problem

Consider the singular

Existence and uniqueness of the IMEX GLM numerical solution on singular perturbation problem

Schneider [35] performed a detailed singular perturbation analysis of implicit (non-partitioned) GLMs. Other authors also studied implicit (non-partitioned) GLMs applied to differential-algebraic problems. Chartier [15] studied the convergence of GLMs for DAEs of index-1, and of stiffly accurate GLMs for DAEs of index-2. Butcher and Chartier [10] developed parallel implicit GLMs and applied them to stiff ordinary differential and differential algebraic equations of index two and three. Schulz

Convergence analysis for index-1 differential-algebraic problems

Schneider [35] performed a detailed singular perturbation analysis of implicit (non-partitioned) GLMs. He provided convergence results for index-1 DAEs, and for index-2 DAEs for a special class of GLMs. Here, and in the next section, we extend his analysis to IMEX GLMs.

Solution expansions

We now formally expand the numerical solutions of method (33a), (33b), (33c), (33d) in series of ε:X[n]=k0X[n],kεk,Z[n]=k0Z[n],kεk,x[n]=k0x[n],kεk,z[n]=k0z[n],kεk. We denote the global errors for each internal and external stage coefficients by:

We insert (53) into (33a), (33b) and expand the function in Taylor series about the O(1) terms to obtain: where the function derivatives are applied to the following multinomial arguments:(Wα,β[n])(,m1X[n],mεm,α times,,1Z[n],ε,β

Discussion

This paper develops new fixed-step convergence results for implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties.

The classical convergence analysis reveals that IMEX GLMs in the class of interest converge with the full theoretical order for sufficiently small step sizes. The upper bound for the step size depends on the method coefficients and on non-stiff Lipschitz constants, but is

Acknowledgements

This work has been supported in part by NSF through awards NSF ACI–1709727 and NSF CCF–1613905, and by the Computational Science Laboratory at Virginia Tech.

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