A nine-stage multi-derivative Runge–Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y′ = f(x, y), y(x0) = y0. The method uses y′ and higher derivatives y(2) to y(6) as in Taylor methods and is combined with a 9-stage Runge–Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor e...

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 ...

A quadrature free, Runge–Kutta discontinuous Galerkin method (QF-RK-DGM) is developed to solve the level set equation written in a conservative form on twoand tri-dimensional unstructured grids. We show that the DGM implementation of the level set approach brings a lot of additional benefits as compared to traditional ENO level set realizations. Some examples of computations are provided that d...

in this paper, the numerical algorithms for solving ‘fuzzy ordinary differential equations’ are considered. a scheme based on the 4th order runge-kutta method is discussed in detail and it is followed by a complete error analysis. the algorithm is illustrated by solving some linear and nonlinear fuzzy cauchy problems.

In this paper, we propose a functional fitting s-stage Runge-Kutta method which is based on the exact integration of the set of the linearly independent functions u i (t), (i = 1,. .. , s). The method is exact when the solution of the ODE can be expressed as the linear combination of u i (t), although the method has an error for general ODE. In this work we investigate the order of accuracy of ...

We propose IMEX HDG-DG schemes for planar and spherical shallow water systems. Of interest is subcritical flow, where the speed of the gravity wave is faster than that of nonlinear advection. In order to simulate these flows efficiently, we split the governing system into a stiff part describing the gravity wave and a non-stiff part associated with nonlinear advection. The former is discretized...

In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multi-core architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on Integral Deferred Correction (IDC), which was itself motivated by Spectral Deferred Correction by Dutt, Greengard and Rokhlin (BIT-2000). The ...