Continuous two-step Runge–Kutta methods for ordinary differential equations

New classes of continuous two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are derived. These methods are developed imposing some interpolation and collocation conditions, in order to obtain desirable stability properties such as A-stability and L-stability. Particular structures of the stability polynomial are also investigated.     

Delay-dependent stability of high order Runge–Kutta methods

This paper is concerned with the study of the delay-dependent stability of Runge–Kutta methods for delay differential equations. First, a new sufficient and necessary condition is given for the asymptotic stability of analytical solution. Then, based on this condition, we establish a relationship between τ(0)-stability and the boundary locus of the stability region of numerical methods for ordinary differential equations. Consequently, a class of high order Runge–Kutta methods are proved to be τ(0)-stable. In particular, the τ(0)-stability of the Radau IIA methods is proved.     

Efficient exponential Runge–Kutta methods of high order: construction and implementation

BIT Numerical Mathematics - Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly...     

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

We derive a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge–Kutta Collocation (RK-C) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate one-degree higher continuous reconstructions of the approximate solutions and pointwise error representations. The reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time-step; they do not hinge on asymptotics.     

Construction of Runge–Kutta methods of Crouch–Grossman type of high order

An approach is described to the numerical solution of order conditions for Runge–Kutta methods whose solutions evolve on a given manifold. This approach is based on least squares minimization using the Levenberg–Marquardt algorithm. Methods of order four and five are constructed and numerical experiments are presented which confirm that the derived methods have the expected order of accuracy.     

Runge–Kutta methods and Banach algebras

The equations defining both the exact and the computed solution to an initial value problem are related to a single functional equation, which can be regarded as prototypical. The functional equation can be solved in terms of a formal Taylor series, which can also be generated using an iteration process. This leads to the formal Taylor expansions of the solution and approximate solutions to initial value problems. The usual formulation, using rooted trees, can be modified to allow for linear combinations of trees, and this gives an insight into the nature of order conditions for explicit Runge–Kutta methods. A short derivation of the family of fourth order methods with four stages is given.     

Explicit Runge–Kutta pairs with lower stage-order

Explicit Runge–Kutta pairs of methods of successive orders of accuracy provide effective algorithms for approximating solutions to nonstiff initial value problems. For each explicit RK method of order of accuracy p, there is a minimum number s _p of derivative evaluations required for each step propagating the numerical solution. For p ≤ 8, Butcher has established exact values of s _p , and for p > 8, his work establishes lower bounds; otherwise, upper bounds are established by various published methods. Recently, Khashin has derived some new methods numerically, and shown that the known upper bound on s ₉ for methods of order p = 9 can be reduced from 15 to 13. His results motivate this attempt to identify parametrically exact representations for coefficients of such methods. New pairs of methods of orders 5,6 and 6,7 are characterized in terms of several arbitrary parameters. This approach, modified from an earlier one, increases the known spectrum of types of RK pairs and their derivations, may lead to the derivation of new RK pairs of higher-order, and possibly to other types of explicit algorithms within the class of general linear methods.     

Explicit pseudo two-step exponential Runge–Kutta methods for the numerical integration of first-order differential equations

Numerical Algorithms - This paper is devoted to the explicit pseudo two-step exponential Runge–Kutta (EPTSERK) methods for the numerical integration of first-order ordinary differential...     

Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods

We present new symmetric fourth and sixth-order symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. We studied compositions using several extra stages, optimising the efficiency. An effective error, E_f, is defined and an extensive search is carried out using the extra parameters. The new methods have smaller values of E_f than other methods found in the literature. When applied to several examples they perform up to two orders of magnitude better than previously known method, which is in very good agreement with the values of E_f.     

Runge–Kutta methods and viscous wave equations

We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.      

Diagonally drift-implicit Runge–Kutta methods of weak order one and two for Itô SDEs and stability analysis

The class of stochastic Runge–Kutta methods for stochastic differential equations due to Rößler is considered. Coefficient families of diagonally drift-implicit stochastic Runge–Kutta (DDISRK) methods of weak order one and two are calculated. Their asymptotic stability as well as mean-square stability (MS-stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.     

Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, a class of nonstandard 2-stage Runge–Kutta methods of order two (we call it nonstandard RK2) is considered. The preservation of some qualitative properties by this class of methods are discussed. In order to illustrate our results, we provide some numerical examples.     

Numerical solution of stochastic differential equations by second order Runge–Kutta methods

In this paper we propose the numerical solutions of stochastic initial value problems via random Runge–Kutta methods of the second order and mean square convergence of these methods is proved. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced. Expectation and variance of the approximating process are computed. Numerical examples show that the approximate solutions have a good degree of accuracy.     

Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation du/dt=λu(t)+μu(t-τ)$\mathrm{d}u/\mathrm{d}t=\lambda u(t)+\mu u(t-\tau )$, where τ$\tau$ is a constant delay and λ,μ$\lambda ,\mu$ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.     

Runge–Kutta methods for jump–diffusion differential equations

In this paper we consider Runge–Kutta methods for jump–diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge–Kutta methods. First, we analyse schemes where the drift is approximated by a Runge–Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge–Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge–Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.     

A-stable Runge–Kutta methods for semilinear evolution equations

We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale. Under the same assumptions, we prove that a class of implicit, A-stable Runge–Kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale. Under the additional assumption that the linear part of the evolution equation is normal or sectorial, we prove full order convergence of the semidiscretization in time for initial data on open sets. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrödinger equation.