Runge-Kutta Stability Research Articles

On the construction of diagonally implicit two–step peer methods with RK stability

In this paper, diagonally implicit two–step peer methods for the numerical solution of initial value problems of order ordinary differential are divided into four types including the combination of explicit and implicit methods in a sequential or parallel environments. In this class of the methods, construction of implicit methods equipped with Runge–Kutta stability property together with A– or L–stability are investigated and examples of such methods are given up to order five. Finally, the efficiency and accuracy of the proposed methods are verified by applying them on some well–known stiff problems.

Self starting general linear methods with Runge–Kutta stability

Self starting general linear methods with Runge–Kutta stability

Optimal explicit Runge–Kutta time stepping for density-based finite-volume solvers

In this paper, we generate optimal Runge–Kutta stability polynomials for several finite-volume spatial discretizations. From these stability polynomials we generate Butcher tableaus, and compare the accuracy and efficiency of these new schemes to commonly used existing classical temporal schemes. Theoretical analysis shows that these optimal schemes are expected to be significantly more efficient, allowing larger time-step sizes to be used relative to the number of Runge–Kutta stages. These predictions are then verified in practice for several test cases including an isentropic vortex, Taylor–Green vortex, and a T106A turbine cascade. Significant performance improvements were observed for all of these cases, with negligible impact on solution accuracy. Importantly, these improvements are achieved with only minor modifications to the solver.

On the implementation of explicit two-step peer methods with Runge–Kutta stability

Explicit general linear method of two-step peer-type of order 1 to 4 suitable for solving non-stiff ordinary differential equations are derived. Estimates of the local truncation error are derived with which a variable stepsize and variable order scheme is implemented. The new methods are compared in numerical experiments with ode45 from Matlab.

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.

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.

Extrapolation of General Linear Methods with Inherent Runge-Kutta Stability

General linear methods have been proven to be very efficient in solving stiff and non-stiff differential equations. Extrapolation is proven to increase the accuracy of any methods. This paper investigates the accuracy and efficiencies of explicit general linear methods with inherent Runge-Kutta stability (IRKs) with and without extrapolation. The numerical results on the Van der Pol (VDP) and Brusselator (Bruss) non-stiff test equations showed that IRKs with extrapolation are more efficient and accurate than itself without extrapolation.

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.

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.

Strong stability preserving implicit–explicit transformed general linear methods

We consider the class of implicit–explicit (IMEX) general linear methods (GLMs) to construct methods where the explicit part has strong stability preserving (SSP) property, while the implicit part of the method has inherent Runge–Kutta stability (IRKS) property, and it is A-, or L-stable. We will also investigate the absolute stability of these methods when the implicit and explicit parts interact with each other. In particular, 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(α)-stable for α∈[0,π∕2]. Finally we furnish examples of SSP IMEX GLMs up to the order p=4 and stage order q=p with optimal SSP coefficients.

EXPLICIT GENERAL LINEAR METHODS WITH A LARGE STABILITY REGION FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, we describe the construction of a class of methods with a large area of the stability region for solving Volterra integro-differential equations. In the structure of these methods which is based on a subclass of explicit general linear methods with and without Runge-Kutta stability property, we use an adequate quadrature rule to approximate the integral term of the equation. The free parameters of the methods are used to obtain methods with a large stability region. The efficiency of the proposed methods is verified with some numerical experiments and comparisons with other existing methods.

Extrapolation of GLMs with IRKS Property to Solve the Ordinary Differential Equations

The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this paper, we investigate the efficiency of extrapolation of explicit general linear methods with Inherent Runge-Kutta stability in solving the non-stiff problems. The numerical experiments are shown for Van der Pol and Brusselator test problems to determine the efficiency of the explicit general linear methods with extrapolation technique. The numerical results showed that method with extrapolation is efficient than method without extrapolation.

Towards a code for nonstiff differential systems based on general linear methods with inherent Runge–Kutta stability

Various issues are discussed which are relevant to the development of the code for nonstiff differential systems based on a class of general linear methods with inherent Runge–Kutta stability.

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.

Strong Stability Preserving General Linear Methods with Runge–Kutta Stability

We investigate strong stability preserving (SSP) general linear methods (GLMs) for systems of ordinary differential equations. Such methods are obtained by the solution of the minimization problems with nonlinear inequality constrains, corresponding to the SSP property of these methods, and equality constrains, corresponding to order and stage order conditions. These minimization problems were solved by the sequential quadratic programming algorithm implemented in MATLAB $$^{\circledR }$$ subroutine fmincon.m starting with many random guesses. Examples of transformed SSP GLMs of order $$p = 1, 2, 3$$ , and 4, and stage order $$q = p$$ have been determined, and suitable starting and finishing procedures have been constructed. The numerical experiments performed on a set of test problems have shown that transformed SSP GLMs constructed in this paper are more accurate than transformed SSP DIMSIMs and SSP Runge–Kutta methods of the same order.

Order reduction phenomenon for general linear methods

The order reduction phenomenon for general linear methods (GLMs) for stiff differential equations is investigated. It turns out that, similarly as for standard Runge–Kutta methods, the effective order of convergence for a large class of GLMs applied to stiff differential systems, is equal to the stage order of the method. In particular, it is demonstrated that the global error ‖e[n]‖ of GLMs of order p and stage order q applied to the Prothero–Robinson test problem y′(t)=λ(y(t)−φ(t))+φ′(t), t∈[t0,T], y(t0)=φ(t0), is O(hq)+O(hp) as h→0 and hλ→−∞. Moreover, for GLMs with Runge–Kutta stability which are A(0)-stable and for which the stability function R(z) of the underlying Runge–Kutta methods, (i.e., the corresponding RK methods which have the same absolute stability properties as the GLMs), is such that R(∞)≠1, the global error satisfies ‖e[n]‖=O(hq+1)+O(hp) as h→0 and hλ→−∞. These results are confirmed by numerical experiments.

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.

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 general linear methods with inherent Runge–Kutta stability

In this paper, we find some relationships among the coefficients matrices of second derivative general linear methods (SGLMs) which are sufficient conditions, but not necessary, to ensure the methods have Runge---Kutta stability (RKS) property. Considering these conditions, we construct some A--- and L---stable SGLMs with inherent RKS of orders up to five. Also, some numerical experiments for the constructed methods in variable stepsize environment are given.