We consider the development of Implicit-Explicit (IMEX) RungeKutta schemes for hyperbolic and kinetic equations in the diffusion limit. In such regime the system relaxes towards a parabolic diffusion equation and it is desirable to have a method that is able to capture the asymptotic behavior with an implicit treatment of limiting diffusive terms. To this goal we reformulate the problem by prop...

In this paper, we consider construct high order semi-implicit integrators using integral deferred correction (IDC) to solve stiff initial value problems. The general framework for the construction of these semi-implicit methods uses uniformly distributed nodes and additive RungeKutta (ARK) integrators as base schemes inside an IDC framework, which we refer to as IDC-ARK methods. We establish un...

The coupled mode theory (CMT) is used to analyze uniform Fiber Bragg gratings. The multi-mode CMT is expressed as the first-order vector ordinary differential equations (ODEs) with coefficients depending on the propagation distance. We show in this paper that by changing variables, the original couple mode equations (CMEs) can be re-casted as constant coefficient ODEs. The eigenvalue and eigenv...

Water pollution is a major global problem that requires ongoing evaluation and revision of water resource policies at all levels, in order to create a healthy living environment. Differential equations are an effective way to analyze such situations. In this paper a system of linear equations with interconnecting pipes is considered for analyzing the pollution of system of lakes through differe...

Automatic differentiation is used for the solution of optimal control problems. Original problem is reduced to a nonlinear programming problem using general RungeKutta integration formulas. Canonical formulas which use a fast automatic differentiation strategy are given to compute derivatives of goal function.

We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method dG(q) and the corresponding implicit RungeKutta-Radau method IRK-R(q) of arbitrary order q ≥ 0 for both linear and nonlinear evolution problems of the form u′ + F(u) = f , with γ-angle bounded operator F. The key ingredient is a novel higher order reconstruction Û of the discrete solution U , whi...

General multi-dimensional autonomous dynamical systems and their numerical discretizations are considered. Nonstandard stability-preserving finite-difference schemes based on the θ-methods and the second-order RungeKutta methods are designed and analyzed. Their elementary stability is established theoretically and is also supported by a set of numerical examples.

A compressible non-body conformal grid method is presented. The method is designed to simulate a large variety of subsonic compressible flows with complex geometries. A hybrid implicit-explicit timediscretization scheme is chosen. The diagonal viscous terms are treated implicitly and all other terms including the convective terms and cross-terms are treated explicitly using a low-storage, 3-Ord...

This is the fth paper in a series in which we construct and study the so-called RungeKutta Discontinuous Galerkin method for numerically solving hyperbolic conservation laws. In this paper, we extend the method to multidimensional nonlinear systems of conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation issues such as the nu...

An approach is described to the numerical solution of order conditions for RungeKutta methods whose solutions evolve on a given manifold. This approach is based on least squares minimization using the Levenberg-Marquardt algorithm. Examples of methods of order four, ve and six are given and numerical experiments are presented which con rm that the derived methods have the expected order of accu...