Diagonally Implicit Multistage Integration Methods Research Articles

ACTIVE EXTRAPOLATION OF DIMSIMS IN NORDSIECK REPRESENTATION

Diagonally implicit multistage integration methods (DIMSIMs) are widely utilized in finding the solution to any problems in the subject of ordinary differential equations. These methods are selected from the general linear methods, which is considerable potential for efficient implementations. The extrapolation is derived from the stability of the explicit Runge-Kutta methods. In this paper, the combination of DIMSIMs with Richardson extrapolation of different orders shows that numerical solutions give higher accuracy when the extrapolation is applied with the base method.

Parallel Implicit-Explicit General Linear Methods

High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) offers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly. This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The first approach is based on diagonally implicit multistage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge-Kutta methods.

Transformed implicit-explicit second derivative diagonally implicit multistage integration methods with strong stability preserving explicit part

In this paper, we discuss the construction of a class of implicit-explicit (IMEX) methods for systems of ordinary differential equations which their right hand side can be split into two parts; nonstiff or mildly stiff part and stiff part. The proposed methods treat the non-stiff part by an explicit second derivative diagonally implicit multistage integration method (SDIMSIM) and the stiff part by an implicit diagonally implicit multistage integration method (DIMSIM). The explicit part of these methods has strong stability preserving (SSP) property and the implicit part is A- and L-stable. We will construct methods with p=q=r=s and p=q+1=r=s up to order four with large SSP coefficients with respect to the large region of absolute stability, assuming that the implicit part of the method has Runge–Kutta stability (RKS) property together with A- and L-stability. These methods are tested on the linear advection-diffusion, advection-reaction and nonlinear shallow water equations, and the numerical results are presented conforming the efficiency and order of constructed methods.

Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part

For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems can be efficiently treated by a class of implicit-explicit (IMEX) diagonally implicit multistage integration methods (DIMSIMs), where the stiff part is integrated by an implicit formula, and the non-stiff part is integrated by an explicit formula. We will construct methods where the explicit part has strong stability preserving (SSP) property, and the implicit part of the method is A-, or L-stable. We will also investigate stability of these methods when the implicit and explicit parts interact with each other. To be more precise, we will monitor the size of the region of absolute stability of the IMEX scheme, assuming that the implicit part of the method is A-, or L-stable. Finally, we furnish examples of SSP IMEX DIMSIMs up to the order four with good stability properties.

Strong stability preserving transformed DIMSIMs

In this paper we investigate the strong stability preserving (SSP) property of transformed diagonally implicit multistage integration methods (DIMSIMs). Within this class, examples of SSP methods of order p = 1 , 2 , 3 and 4 and stage order q = p have been determined, and also suitable starting and finishing procedure have been constructed. The numerical experiments performed on a set of test problems have shown that SSP transformed DIMSIMs can be more accurate and competitive with SSP Runge–Kutta methods of the same order.

Construction of IMEX DIMSIMs of high order and stage order

For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems can be efficiently treated by a class of implicit–explicit (IMEX) diagonally implicit multistage integration methods (DIMSIMs), where the stiff part is integrated by implicit formula, and the non-stiff part is integrated by an explicit formula. We analyze stability of these methods when the implicit and explicit parts interact with each other. We look for methods with large absolute stability region, assuming that the implicit part of the method is A(α)-, A-, or L-stable. Finally, we furnish examples of IMEX DIMSIMs of order p=5 and p=6 and stage order q=p, with good stability properties. Numerical examples illustrate that the IMEX schemes constructed in this paper do not suffer from order reduction phenomenon for some range of stepsizes.

A new approach to the construction of DIMSIMs of high order and stage order

In this work we describe a new approach to the construction of diagonally implicit multistage integration methods (DIMSIMs) for the numerical solution of initial value problems for ordinary differential equations (ODEs). Differential Evolution, a very popular computational intelligence technique is employed to construct type 1 and type 2 methods with better or equivalent characteristics to the methods presented in the literature. The numerical results in selected problems justify this argument.

Partitioned and Implicit–Explicit General Linear Methods for Ordinary Differential Equations

Implicit–explicit (IMEX) time stepping methods can efficiently solve differential equations with both stiff and nonstiff components. IMEX Runge–Kutta methods and IMEX linear multistep methods have been studied in the literature. In this paper we study new implicit–explicit methods of general linear type. We develop an order conditions theory for high stage order partitioned general linear methods (GLMs) that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms. Next, we propose a family of IMEX schemes based on diagonally-implicit multi-stage integration methods and construct practical schemes of order up to three. Numerical results confirm the theoretical findings.

A Second-order Diagonally-Implicit-Explicit Multi-Stage Integration Method

Implicit-explicit time stepping methods have proved useful for effciently solving problems with both still and nonstill components. IMEX Runge-Kutta methods and IMEX multistep methods have been studied in the literature. But there are no available IMEX general linear methods (IMEX-GLMs). In this paper, we construct IMEX schemes based on diagonally-implicit multi-stage integration methods. The new algorithms have great potential for practical use. Numerical results indicate the observed order of accuracy matches the theoretical order.

A reliable error estimation for diagonally implicit multistage integration methods

A new approach to the estimation of the local discretization error for diagonally implicit multistage integration methods (DIMSIMs) is described. The error estimates that are obtained are very accurate and very reliable for both explicit and implicit methods for any stepsize pattern.

The construction of order 4 DIMSIMs for ordinary differential equations

Diagonally Implicit Multistage Integration Methods (DIMSIMs) of type 1 and 2 have considerable potential as numerical algorithms for ordinary differential equations. The aim of this paper is to construct such methods of order 4 of type 1 and 2, which completes the set for orders 1–8.

Implementation of Diagonally Implicit Multistage Integration Methods for Ordinary Differential Equations

We investigate the implementation of diagonally implicit multistage integration methods (DIMSIMs). The implementation issues addressed are the local error estimation, changing stepsize using the Nordsieck technique, and the construction of continuous interpolants. Numerical experiments with a method of order three indicate that the error estimates that have been constructed are very reliable in both a fixed and a variable stepsize environment.

Regularity properties of multistage integration methods

The numerical method for ordinary differential equations is regular if it has the same set of finite asymptotic values as the underlying differential system. This paper examines the regularity and strong regularity properties of diagonally implicit multistage integration methods (DIMSIMs) introduced recently by J.C. Butcher. A sufficient condition for regularity and strong regularity of such methods of any order is given and it is proved that this condition is also necessary for two-step two-stage DIMSIMs of order greater than or equal to two. It is also demonstrated that there exist regular schemes in the class of explicit DIMSIMs. This is in contrast to explicit Runge-Kutta methods with more than one stage, which are always irregular.

Nordsieck representation of DIMSIMs

A new representation for diagonally implicit multistage integration methods (DIMSIMs) is derived in which the vector of external stages directly approximates the Nordsieck vector. The methods in this formulation are zero-stable for any choice of variable mesh. They are also easy to implement since changing step-size corresponds to a simple rescaling of the vector of external approximations. The paper contains an analysis of local truncation error and of error accumulation in a variable step-size situation.

General linear methods with external stages of different orders

The approach introduced recently by Albrecht to derive order conditions for Runge-Kutta formulas based on the theory of A-methods is also very powerful for the general linear methods. In this paper, using Albrecht's approach, we formulate the general theory of order conditions for a class of general linear methods where the components of the propagating vector of approximations to the solution have different orders. Using this theory we derive a class of diagonally implicit multistage integration methods (DIMSIMs) for which the global order is equal to the local order. We also derive a class of general linear methods with two nodal approximations of different orders which facilitate local error estimation. Our theory also applies to the class of two-step Runge-Kutta introduced recently by Jackiewicz and Tracogna.

Construction of diagonally implicit general linear methods of type 1 and 2 for ordinary differential equations

We describe the construction of diagonally implicit multistage integration methods (DIMSIMs) of type 1 and 2 with the same stability properties as explicit Runge-Kutta methods or implicit SDIRK methods, respectively, of appropriate order. Such methods are intended for the numerical integration of nonstiff or stiff differential systems in a sequential computing environment. Examples of pqrst DIMSIMs are given with p, q, r, s ≤ 4 and t = 1 or 2, where p is the order, q is the stage order, r is the number of external stages, s is the number of internal stages, and t is the type of the method. Coefficients of the methods of order 4 were obtained numerically with the aid of continuation programs from PITCON, ALCON, and HOMPACK.

A transformation for the analysis of DIMSIMs

The special case of diagonally-implicit multistage integration methods is considered in which the order, the stage order, the number of values passed between steps and the number of stages in a step, all coincide. It is shown that a similarity transformation can be applied to the matrices characterizing the method so as to simplify the expression for the stability polynomial and thus aid in the search for methods with acceptable stability.

Diagonally-implicit multi-stage integration methods

Even though general linear methods were proposed as long as 25 years ago, they have been widely adopted as practical methods for incorporation into numerical software. The aim of the presented paper is to select from the large family of possible general linear methods, just a single class which has cosiderable potential for efficient implementation. The class, which will be known by the acronym DIMSIM, divides into four special subclasses. These have possible applications depending on the stiff or nonstiff nature of a problem to be solved and the parallel or sequential nature of the computing environment in which the solution is to be performed.