On the iterated Crank-Nicolson for hyperbolic and parabolic equations in numerical relativity

The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used in numerical relativity for the solution of both hyperbolic and parabolic partial differential equations. We here extend the recent work on the stability of this scheme for hyperbolic equations by investigating the properties when the average between the predicted and corrected values is made with unequal weights and when the scheme is applied to a parabolic equation. We also propose a variant of the scheme in which the coefficients in the averages are swapped between two corrections leading to systematically larger amplification factors and to a smaller numerical dispersion.Comment: 7 pages, 3 figure

Unconditional Stability for Multistep ImEx Schemes: Theory

This paper presents a new class of high order linear ImEx multistep schemes with large regions of unconditional stability. Unconditional stability is a desirable property of a time stepping scheme, as it allows the choice of time step solely based on accuracy considerations. Of particular interest are problems for which both the implicit and explicit parts of the ImEx splitting are stiff. Such splittings can arise, for example, in variable-coefficient problems, or the incompressible Navier-Stokes equations. To characterize the new ImEx schemes, an unconditional stability region is introduced, which plays a role analogous to that of the stability region in conventional multistep methods. Moreover, computable quantities (such as a numerical range) are provided that guarantee an unconditionally stable scheme for a proposed implicit-explicit matrix splitting. The new approach is illustrated with several examples. Coefficients of the new schemes up to fifth order are provided.Comment: 33 pages, 7 figure

Status of research at the Institute for Computer Applications in Science and Engineering (ICASE)

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science is summarized

A Parallel Algorithm for Solving the Advection Equation with a Retarded Argument

We describe a parallel implementation of a difference scheme for the advection equation with time delay on a hybrid architecture computation system. The difference scheme has the second order in space and the first order in time and is unconditionally stable. Performance of a sequential algorithm and several parallel implementations with the MPI technology in the C++ language has been studied

Non-Standard Discretization of the Advection-Diffusion-Reaction Equation with Logistic Growth Reaction

The goal of this work is to make a comparative analysis between the standard finite difference method and the non-standard finite difference method, then to make a non-standard discretization of the advection-diffusion-reaction equation with a reaction modelling a logistic growth which can be the evolution of the concentration of a microbial population in a medium, the equation will thus model transport and diffusion of this population in the aforementioned medium in one dimension of space and one makes numerical simulations to compare the non-standard scheme and the Euler’s scheme, explicit in time, implicit for the first order derivative in  and centered for the second order derivative in . One arrives by constructing a scheme of the advection-reaction equation, then adds the term of diffusion to obtain the non-standard scheme

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.IS