Large Regions Of Absolute Stability Research Articles

A class of explicit second derivative general linear methods for non-stiff ODEs

In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.

An optimized hybrid block method for solving nonlinear singularly perturbed problems using non-uniform Jain-type meshes

This work aims to implement a one-step block method with three optimized hybrid points for solving singularly perturbed nonlinear initial value problems on non-uniform meshes. The basic properties of the method are investigated and it is shown that the proposed method is consistent, zero-stable, convergent and has a large region of absolute stability. The presented numerical results confirm the efficiency and accuracy of the proposed method.

Explicit Nordsieck second derivative general linear methods for ODEs

The paper deals with the construction of explicit Nordsieck second derivative general linear methods with \(s\) stages of order \(p\) with \(p=s\) and high stage order \(q=p\) with inherent Runge–Kutta or quadratic stability properties. Satisfying the order and stage order conditions together with inherent stability conditions leads to methods with some free parameters, which will be used to obtain methods with a large region of absolute stability. Examples of methods with \(r\) external stages and \(p=q=s=r-1\) up to order five are given, and numerical experiments in a fixed stepsize environment are presented. doi: https://doi.org/10.1017/S1446181122000049

Implicit-explicit second derivative general linear methods with strong stability preserving explicit part

In this paper, we discuss the class of implicit–explicit (IMEX) methods for systems of ordinary differential equations where the explicit part has strong stability preserving (SSP) property, while the implicit part of the method has Runge–Kutta stability property together with A-, or L-stability. The explicit part of these methods is treated by the explicit second derivative general linear methods and the implicit part is treated by the implicit general linear methods. We construct methods of order p=s and stage order q=p and q=p−1 with r=2, of order p=q=s−1=r−1 and of order p=q+1=s−1=r−1 up to the order p=4 with large SSP coefficients with respect to the large region of absolute stability when the implicit and explicit parts interact with each other. In the case of methods with p=q=r−1=s−1 and p=q+1=s−1=r−1, the implicit part has inherent Runge–Kutta stability property, and is L-stable. Finally, we test the proposed SSP IMEX schemes on some one dimensional linear and nonlinear problems, and the results for the expected order of convergence are presented.

EXPLICIT NORDSIECK SECOND DERIVATIVE GENERAL LINEAR METHODS FOR ODES

AbstractThe paper deals with the construction of explicit Nordsieck second derivative general linear methods with s stages of order p with $p=s$ and high stage order $q=p$ with inherent Runge–Kutta or quadratic stability properties. Satisfying the order and stage order conditions together with inherent stability conditions leads to methods with some free parameters, which will be used to obtain methods with a large region of absolute stability. Examples of methods with r external stages and $p=q=s=r-1$ up to order five are given, and numerical experiments in a fixed stepsize environment are presented.

High order extended boundary value methods for the solution of stiff systems of ODEs

A class of high order extended boundary value methods (HEBVMs) suitable for the numerical approximation of stiff systems of ordinary differential equations (ODEs) is constructed. This class of BVMs is based on the second derivative class of linear multistep formulas (LMF) and it provides a set of very highly stable methods that can produce considerably accurate solutions to stiff systems whose Jacobians have some large eigenvalues lying close to the imaginary axis. The class of BVMs derived herein is of high order, small error constants and large region of absolute stability. Specifically, it is Ok1,k2-stable, Ak1,k2-stable with (k1,k2) -boundary conditions and order p=k+4 for values of the step length k≥1. The numerical results obtained from standard linear and non-linear stiff systems indicate that this scheme is highly competitive with existing methods.

A new class of strong stability preserving general linear methods

We systematically investigate strong stability preserving general linear methods of order p, stage order q=p or q=p−1, with r=p+1 external approximations, and s=p−1 internal approximations, for numerical solution of differential systems. Examples of methods of order p and stage order q=p or q=p−1, with large strong stability preserving coefficients and large regions of absolute stability, are provided for p=2, p=3, and p=4. The results of numerical experiments confirm that the methods constructed in this paper achieve the expected order of accuracy, do not produce spurious oscillations, and they are suitable to preserve the monotonicity of the numerical solution, when applied to discretization of hyperbolic conservation laws with discontinuous initial conditions.

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.

Paired explicit Runge-Kutta schemes for stiff systems of equations

In this paper we introduce a family of explicit Runge-Kutta methods, referred to as Paired Explicit Runge-Kutta (P-ERK) schemes, that are suitable for the solution of stiff systems of equations. The P-ERK approach allows Runge-Kutta schemes with a large number of derivative evaluations and large region of absolute stability to be used in the stiff parts of a domain, while schemes with relatively few derivative evaluations are used in non-stiff parts to reduce computational cost. Importantly, different P-ERK schemes with different numbers of derivative evaluations can be chosen based on local stiffness requirements and seamlessly paired with one another. We then verify that P-ERK schemes obtain their designed order of accuracy using the Euler equations with arbitrary combinations of schemes. We then demonstrate that P-ERK schemes can achieve speedup factors of approximately five for simulations using the Navier-Stokes equations including laminar and turbulent flow over an SD7003 airfoil. These results demonstrate that P-ERK schemes can significantly accelerate the solution of stiff systems of equations when using an explicit approach, and that they maintain accuracy with respect to conventional Runge-Kutta methods and available reference data.

General linear methods with large stability regions for Volterra integral equations

General linear methods are well-known as a large family of methods for the numerical solution of Volterra integral equations of the second kind. This paper is concerned with the construction of such methods with a large region of order p and high stage order $$q=p$$ with a large region of absolute stability. Some numerical results are presented which indicate the effectiveness of the proposed schemes.

Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems

Second derivative Runge-Kutta collocation methods for the numerical solution of sti system of rst order initial value problems in ordinary dierential equations are derived and studied. The inclusion of the second derivative terms enabled us to derive a set of methods which are convergent with large regions of absolute stability. Although the implementation of the methods remains iterative in a precisely dened way, the advantage gained makes them suitable for solving sti system of equations with large Lipschitz constants. The derived methods are illustrated by the applications to some test problems of sti system found in the literature and the numerical results obtained conrm the potential of the second derivative methods.

The numerical integration of stiff systems using stable multistep multiderivative methods

In this paper we describe the construction of stable multistep multiderivative methods designed for continuous numerical integration of stiff systems of initial value problems in ordinary dierential equations. These methods are obtained based on the multistep collocation technique, which are shown to be A-stable, convergent with large regions of absolute stability. They are suitable for solving stiff systems of initial value problems with large eigenvalues lying close to the imaginary axis. Numerical experiments illustrate the behaviour of the methods, which show that they are competitive with stiff integrators that are known to have strong stability characteristic properties. Comparison of the solution curves obtained is in good agreement with the exact solutions which demonstrate the reliability and usefulness of the methods.

Efficient second derivative methods with extended stability regions for non-stiff IVPs

We describe the construction of explicit Nordsieck sequential second derivative general linear methods with a large region of absolute stability. The constructed methods are of order \(p=q=s\), where q and s are the stage order and the number of internal stages, respectively. To compare, we also construct methods in this class which possess Runge–Kutta stability property. Implementation of the constructed methods in a variable stepsize mode is discussed and efficiency of these methods is verified by some numerical experiments in fixed and variable stepsize environment.

Highly stable implicit–explicit Runge–Kutta methods

We investigate implicit–explicit (IMEX) Runge–Kutta (RK) methods for differential systems with non-stiff and stiff processes. The construction of such methods with large regions of absolute stability of the ‘explicit part’ of the method assuming that the ‘implicit part’ of the scheme is A-stable, is described. We also describe the construction of IMEX RK methods, where the ‘explicit part’ of the schemes have strong stability properties. Examples of highly stable IMEX RK methods are provided up to the order p=4. Numerical examples are also given which illustrate good performance of these schemes.

Accurate Implicit–Explicit General Linear Methods with Inherent Runge–Kutta Stability

We investigate implicit---explicit (IMEX) general linear methods (GLMs) with inherent Runge---Kutta stability (IRKS) for differential systems with non-stiff and stiff processes. The construction of such formulas starts with implicit GLMs with IRKS which are A- and L-stable, and then we `remove' implicitness in non-stiff terms by extrapolating unknown stage derivatives by stage derivatives which are already computed by the method. Then we search for IMEX schemes with large regions of absolute stability of the `explicit part' of the method assuming that the `implicit part' of the scheme is $$A(\alpha )$$A(?)-stable for some $$\alpha \in (0,\pi /2]$$??(0,?/2]. Examples of highly stable IMEX GLMs are provided of order $$1\le p\le 4$$1≤p≤4. Numerical examples are also given which illustrate good performance of these schemes.

Second derivative of high-order accuracy methods for the numerical integration of stiff initial value problems

In Mitsui (Sci Eng Rev Doshisha Univ Jpn 51(3):181–190, 2010) the author introduced a new class of discrete variable methods known as look-ahead linear multistep methods (LALMMs) which consist of a pair of predictor-corrector (PC), including a function value at one more step beyond the present step for the numerical solution of ordinary differential equations. Two-step family of Look-Ahead linear multistep methods of fourth-order pair were derived and shown to be A( $$\theta $$ )-stable in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181–188, 2011). The derived integration methods are of low orders and unfortunately cannot cope with stiff systems of ordinary differential equations. In this paper we extend the concept adopted in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181–188, 2011) to construct second-derivative of high-order accuracy methods with off-step points which behave essentially like one-step methods. The resulting integration methods are A-stable, convergent, with large regions of absolute stability, suitable for stiff systems of ordinary differential equations. Numerical comparisons of the new methods have been made and enormous gains in efficiency are achieved.

A New One-Twelfth Step Continuous Block Method for the Solution of Modeled Problems of Ordinary Differential Equations

In this paper, we developed a new continuous block method by the method of interpolation and collocation to derive new scheme. We adopted the use of power series as a basis function for approximate solution. We evaluated at off grid points to get a continuous hybrid multistep method. The continuous hybrid multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent, zero stable and convergent. The results were found to compete favorably with the existing methods in terms of accuracy and error bound. In particular, the scheme was found to have a large region of absolute stability. The new method was tested on real life problem namely: Dynamic model.

Theoretical and Numerical Analysis of a Class of Semi-Implicit Semi-Lagrangian Schemes Potentially Applicable to Atmospheric Models

Abstract In this paper, theoretical and numerical analyses of the properties of some complex semi-Lagrangian methods are performed to deal with the issues of the instability associated with the treatment of the nonlinear part of the forcing term. A class of semi-Lagrangian semi-implicit schemes is proposed using a modified TR-BDF2 method, which is the combination of the trapezoidal rule (TR) and the second-order backward differentiation formula (BDF2). The process used for the nonlinear term includes two stages as predictor and corrector in the trapezoidal method and one stage for the BDF2 method. For the treatment of the linear term, the implicit trapezoidal method is employed in the first step, the explicit trapezoidal method in the second step, and the implicit BDF2 method in the third step. The combination of these techniques leads to a family of schemes that has a large region of absolute stability, performs well for the purely oscillatory cases, and has good qualities in terms of accuracy and convergence. The use of the explicit method for the linear term in the second step makes the proposed class of schemes competitive in terms of efficiency compared to some well-known schemes that use two steps. Numerical experiments presented herein confirm that the proposed class of schemes performs well in terms of stability, accuracy, convergence, and efficiency in comparison with other, previously known, semi-Lagrangian semi-implicit schemes and semi-implicit predictor–corrector methods. The potential practical application of the proposed class of schemes to a weather prediction model or any other atmospheric model is not discussed and could be the subject of other forthcoming studies.

HIGH ORDER BLOCK IMPLICIT MULTI-STEP (HOBIM) METHODS FOR THE SOLUTION OF STIFF ORDINARY DIFFERENTIAL EQUATIONS

The search for higher order A-stable linear multi-step methods has been the interest of many numerical analyst and has been realized through either higher derivatives of the solution or by inserting additional off step points,supper future points and the likes.These methods are suitable for the solution of stiff differential equations which exhibit characteristics that place severe restriction on the choice of step size. It becomes necessary that only methods with large regions of absolute stability remain suitable for such equa- tions. In this paper, high order block implicit multi-step methods of the hybrid form up to order twelve have been constructed using the multi-step collocation approach by inserting one or more off step points in the multi-step method. The accuracy and stability properties of the new methods are investigated and are shown to yield A- stable methods, a property desirable of methods suitable for the solution of stiff ODEs. The new High Order Block Implicit Multistep methods used as block integrators are tested on stiff differential systems and the results reveal that the new methods are efficient and compete favorably with the state of the art Matlab ode23 code.

Two-step second-derivative high-order methods with two off-step points for solution of stiff systems

Two-Step Second-Derivative High-Order Methods with off-step points suitable for the approximate numerical integration of stiff systems of first order initial value problems in ordinary differential equations are developed. The second derivative terms in the methods provide more freedom to derive a set of methods which are highly stable, convergent, with large regions of absolute stability. Although the cost of calculating the second derivatives may be higher than the first derivatives, the advantage gained makes them suitable for solving stable systems with large Lipschitz constants. Another requirement demands that the second derivative of high-order methods are to be solved iteratively rather than directly in the usual way of the linear multistep methods, however, efficient methods for doing this is provided from the associated continuous scheme of each method, by the evaluation of the continuous scheme at some off-step points. The derived methods are illustrated by the applications to some test problems of stiff systems and the numerical results obtained confirm the accuracy and efficiency of these methods.