We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE’s and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total...

The construction of new explicit Runge–Kutta methods taking into account, not only their accuracy, but also the preservation of Quadratic Invariants (QIs) is studied. An expression of the error of conservation of a QI by a Runge–Kutta method is given, and a new six–stage formula with classical order four and seventh order of QI–conservation is obtained by choosing their coefficients so that the...

We investigate conservative properties of Runge-Kutta methods for Hamiltonian PDEs. It is shown that multi-symplecitic Runge-Kutta methods preserve precisely norm square conservation law. Based on the study of accuracy of Runge-Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy of conservation laws of energy and momentum for H...

This paper presents an overview of high-order implicit time integration methods and their associated properties with a specific focus on their application to computational fluid dynamics. A framework is constructed for the development and optimization of general implicit time integration methods, specifically including linear multistep, Runge-Kutta, and multistep Runge-Kutta methods. The analys...

-A pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step. ...

This paper continues earlier work by the same author concerning the stability and B-convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable, Bor optimally B-convergent ...

This paper deals with the stability of Runge–Kutta methods for a class of stiff systems of nonlinear Volterra delay-integro-differential equations. Two classes of methods are considered: Runge–Kutta methods extended with a compound quadrature rule, and Runge– Kutta methods extended with a Pouzet type quadrature technique. Global and asymptotic stability criteria for both types of methods are de...

We study the stability of Runge-Kutta methods for the time integration of semidiscrete systems associated with time dependent PDEs. These semidiscrete systems amount to large systems of ODEs with the possibility that the matrices involved are far from being normal. The stability question of their Runge-Kutta methods, therefore, cannot be addressed by the familiar scalar arguments of eigenvalues...

An error analysis of Runge-Kutta convolution quadrature is presented for a class of nonsectorial operators whose Laplace transform satisfies, besides the standard assumptions of analyticity in a half-plane Re s > σ0 and a polynomial bound O(s 1) there, the stronger polynomial bound O(s2) in convex sectors of the form | arg s| ≤ π/2 − θ < π/2 for θ > 0. The order of convergence of the Runge-Kutt...