Derivative General Linear Methods 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.

Strong Stability Preserving Second Derivative General Linear Methods Based on Taylor Series Conditions for Discontinuous Galerkin Discretizations

Strong Stability Preserving Second Derivative General Linear Methods Based on Taylor Series Conditions for Discontinuous Galerkin Discretizations

Algebraic stability and irreducibility of second derivative methods

Algebraic stability has been already studied for general linear methods (GLMs) to analyze the behavior of the methods applying to the non-linear problems. We discuss this concept for the class of second derivative GLMs (SGLMs) and derive sufficient conditions on the coefficients of these methods guaranteeing the algebraic stability of the methods. Moreover, by defining irreducible SGLMs, we analyze algebraically stable SGLMs which are also irreducible. Then, we construct some examples of the methods up to order four.

HIGH ORDER SECOND DERIVATIVE DIAGONALLY IMPLICIT MULTISTAGE INTEGRATION METHODS FOR ODES

Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.

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.

Strong stability preserving implicit and implicit–explicit second derivative general linear methods with RK stability

Strong stability preserving implicit and implicit–explicit second derivative general linear methods with RK stability

Composite symmetric second derivative general linear methods for Hamiltonian systems

Composite symmetric second derivative general linear methods for Hamiltonian systems

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.

Variable stepsize SDIMSIMs for ordinary differential equations

Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order p and stage order q=p, some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.

Strong Stability Preserving Second Derivative General Linear Methods with Runge–Kutta Stability

In this paper we describe the construction of second derivative general linear method with Runge–Kutta stability property preserving the strong stability properties of spatial discretizations. Then we present such methods that are obtained by the solution of the constrained minimization problem with nonlinear inequality constraints, corresponding to the strong stability preserving property of these methods, and equality constraints, corresponding to the order, stage order and Runge–Kutta stability conditions. The derived methods are of order $$p=q$$ with $$r=2$$ and $$s=p$$ or $$s=p+1$$ , of order $$p=q=s=r-1$$ and of order $$p=q+1=s=r$$ , where q, s and r are the stage order, the number of internal and the number of external stages, respectively. Efficiency of the proposed methods together with verification of the order of convergence and capability of these methods in solving partial differential equations with smooth and discontinues initial data are shown by some numerical experiments.

Partitioned second derivative methods for separable Hamiltonian problems

In this paper, the partitioned second derivative general linear methods for solving separable Hamiltonian problems are derived which are G-symplectic, interchange symmetry and have zero parasitic growth factors. Order conditions for such methods based on the theory of rooted trees are provided and examples of partitioned pairs of order 2 and 3 are given. The experiments have been performed on some separable problems and no drift in the variation of the Hamiltonian is observed in numerical experiments for long time intervals. Also, the results of numerical experiments demonstrate the efficiency of our new methods in comparison with some symplectic and G-symplectic methods.

Implicit–explicit second derivative diagonally implicit multistage integration methods

We introduce a class of implicit–explicit (IMEX) schemes for the numerical solution of initial value problems of differential equations with both non-stiff and stiff components in which non-stiff and stiff solvers are, respectively, based on the explicit general linear methods (GLMs) and implicit second derivative GLMs (SGLMs). The order conditions of the proposed IMEX schemes are obtained. Linear stability properties of the methods are analyzed and then methods up to order four with a large area of absolute stability region of the pair are constructed assuming that the implicit part of the methods is L-stable. Due to the high stage orders of the constructed methods, they are not marred by order reduction. This is verified by the numerical experiments which demonstrate the efficiency of the proposed methods, too.

Implementation of second derivative general linear methods

In this paper, the implementation of second derivative general linear methods (SGLMs) in a variable stepsize environment using Nordsieck technique is discussed and various implementation issues are investigated. All coefficients of a method of order four together with its error estimate are obtained. The method is derived with the aim of good zero-stability properties for a large range of ratios of sequential stepsizes to implement in a variable stepsize mode. The numerical experiments indicate that the constructed error estimate is very reliable in a variable stepsize environment and beautifully confirm the efficiency and robustness of the proposed scheme based on SGLMs. Moreover, the results verify that the proposed scheme outperforms the code ode15s from Matlab ODE suite on systems whose Jacobian has eigenvalues which are close to the imaginary axis.

Strong stability preserving second derivative diagonally implicit multistage integration methods

Construction of strong stability preserving second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods (SGLMs) with and without quadratic stability (QS) has been investigated. Examples of such methods with p=q=r=s=2,3 and 4 have been constructed, where p is the order, q is the stage order, r is the number of external stages, and s is the number of internal stages of the method. To show the efficiency of the proposed methods together with verification of the order of convergence and capability of these methods in preserving some nonlinear stability properties such as total variation and positivity, several one dimensional numerical experiments are performed.

Order conditions for second derivative general linear methods

Motivated by the extension of Albrecht approach proposed for Runge–Kutta methods and general linear methods, we explain the derivation of order conditions, without any restriction on the stage order, for second derivative general linear methods. Using this approach, we derive general order conditions for second derivative general linear methods up to order six. To illustrate, exact coefficients for several low stage order methods with some desirable stability properties are presented.

Strong Stability Preserving Second Derivative General Linear Methods

In this paper, we find sufficient strong stability preserving (SSP) conditions for second derivative general linear methods (SGLMs). Then we construct some optimal SSP SGLMs of order p up to eight and stage order $$1\le q\le p$$ with two external stages and $$2\le s\le 10$$ internal stages, which have larger effective Courant–Friedrichs–Lewy coefficients than the other existing class of the methods such as two derivative Runge–Kutta methods investigated by Christlieb et al., the class of two-step Runge–Kutta methods introduced by Ketcheson et al., the class of multistep multistage methods studied by Constantinescu and Sandu, and general linear methods investigated by Izzo and Jackiewicz. Some numerical experiments for scalar and systems of equations are provided which demonstrate that the constructed SSP SGLMs achieve the predicted order of convergence and preserve monotonicity.

Projection of Second Derivative Methods for Ordinary Differential Equations with Invariants

For the numerical solution of ordinary differential equations with some invariants, several efficient integrators have been introduced. For such problems with known invariants, we use explicit second derivative general linear methods with standard projection technique that, while preserving invariants of the problems as well, their low computational cost makes them more efficient than the existing conservative integrators. To show the capability and advantages of the proposed schemes, some numerical experiments are presented.

Efficient Nordsieck second derivative general linear methods: construction and implementation

In this paper, by considering the order conditions for second derivative general linear methods of order p and stage order $$q=p-1$$ , we investigate construction and implementation of these methods in the Nordsieck form with $$r=s+1=p$$ , where s and r are the number of internal and external stages of the method, respectively. Constructed methods are A- and L-stable which possess Runge–Kutta stability property. Some numerical experiments are provided in a variable stepsize environment to validate the efficiency of the constructed methods and reliability of the proposed error estimates.

Construction of the Nordsieck second derivative methods with RK stability for stiff ODEs

In this paper, we study the construction and implementation of special Nordsieck second derivative general linear methods of order p and stage order $$q=p$$ in which the number of input and output values is $$r=p$$ rather than $$r=p+1$$ . We will construct A- and L-stable methods of orders three and four in this form with Runge–Kutta stability properties. The efficiency of the constructed methods and reliability of the proposed error estimates are shown by implementing of the methods in a variable stepsize environment on some well-known stiff problems.