The implementation of implicit Runge{Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modiied Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers 5] substitute the Runge{Kutta matrix A in the Newton process for a triangul...

In this paper, we analyze the stability of the fourth order Runge-Kutta method for integrating semi-discrete approximations of time-dependent partial differential equations. Our study focuses on linear problems and covers general semi-bounded spatial discretizations. A counter example is given to show that the classical four-stage fourth order Runge-Kutta method can not preserve the one-step st...

New Runge–Kutta methods for method of lines solution of systems of ordinary differential equations arising from discretizations of spatial derivatives in hyperbolic equations, by Chebyshev or modified Chebyshev methods, are introduced. These Runge–Kutta methods optimize the time step necessary for stable solutions, while holding dispersion and dissipation fixed. It is found that maximizing disp...

We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve th...

In the present paper an attempt is made for the solution of SDE (Stochastic differential equation ) using different numerical simulation . Here the four different technique has been adopt for the two test problem for the verification process . Main emphasis is given on the RKM (Runge kutta Method) in which the solution has minimum number of absolute error .i.e more accurate then other. some of ...

Separable Hamiltonian systems of differential equations have the form dp/dt = -dH/dq, dq/dt = dH/dp, with a Hamiltonian function H that satisfies H = T(p) + K(q) (T and V are respectively the kinetic and potential energies). We study the integration of these systems by means of partitioned Runge-Kutta methods, i.e., by means of methods where different Runge-Kutta tableaux are used for the p and...

Introduction PACT Abstract A parallel implementation for multi-implicit Runge-Kutta methods with real eigen-values is described. The parallel method is analysed and the algorithm is devised. For the problem with d domains, the amount within the s-stage Runge-Kutta method, associated with the solution of system, is proportional to (sd) 3. The proposed parallelisation transforms the above system ...

Abstract. We investigate the strong convergence rate of both Runge–Kutta methods and simplified step-N Euler schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions with H ∈ ( 2 , 1). These two classes of numerical schemes are implementable in the sense that the required information from the driving noises are only their increments. We prove the sol...

The optimization of some W-methods [7] for the time integration of time-dependent PDEs in several spatial variables is considered. In [2, Theorem 1] several three-parametric families of three-stage W-methods for the integration of IVPs in ODEs were studied. Besides, the optimization of several specific methods for PDEs when the Approximate Matrix Factorization Splitting (AMF) [3, 4] is used to ...

in this paper, numerical spline-based differential quadrature is presented for solving the boundary and initial value problems, and its application is used to solve the fixed rectangular membrane vibration equation. for the time integration of the problem, the runge–kutta and spline-based differential quadrature methods have been applied. the runge–kutta method was unstable for solving the prob...