We are concerned with the solution of time-dependent nonlinear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge-Kutta-type time stepping (discretisation in time). The introduced nonlinear blending procedure allows us to retain the explicit character of the time stepping ...

Galerkin fully discrete approximations for parabolic equations with time-dependent coefficients are analyzed. The schemes are based on implicit Runge-Kutta methods, and are coupled with preconditioned iterative methods to approximately solve the resulting systems of linear equations. It is shown that for certain classes of Runge-Kutta methods, the fully discrete equations exhibit parallel featu...

Research on parallel iterated methods based on Runge-Kutta formulas both for stii and non-stii problems has been pioneered by van der Houwen et al., for example see 8, 9, 10, 11]. Burrage and Suhartanto have adopted their ideas and generalized their work to methods based on Multistep Runge-Kutta of Radau type 2] for non-stii problems. In this paper we discuss our methods for stii problems and s...

This paper concerns predictive stepsize control applied to high order methods for temporal discretization in reservoir simulation. The family of Runge-Kutta methods is presented and in particular the explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods are described. A predictive stepsize adjustment rule based on error estimates and convergence control of the integrated iterative so...

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...

Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the is solved numerically. We propose to ensure positivity other by applying Runge-Kutta integration which method weights are adapted order enforce bounds. The chosen at each step after calculating stage derivatives, a way that also pre...

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 ...