On the derivation of explicit two-step peer methods
The so-called two-step peer methods for the numerical solution of Initial Value Problems (IVP) in differential systems were introduced by R. Weiner, B.A. Schmitt and coworkers as a tool to solve different types of IVPs either in sequential or parallel ...
Numerical properties of high order discrete velocity solutions to the BGK kinetic equation
A high order numerical method for the solution of model kinetic equations is proposed. The new method employs discontinuous Galerkin (DG) discretizations in the spatial and velocity variables and Runge-Kutta discretizations in the temporal variable. The ...
Level set based multi-scale methods for large deformation contact problems
We consider the numerical simulation of contact problems in elasticity with large deformations. The non-penetration condition is described by means of a signed distance function to the obstacle's boundary. Techniques from level set methods allow for an ...
Numerical experiments on the condition number of the interpolation matrices for radial basis functions
Through numerical experiments, we examine the condition numbers of the interpolation matrix for many species of radial basis functions (RBFs), mostly on uniform grids. For most RBF species that give infinite order accuracy when interpolating smooth f(x)-...
Exponentially-convergent strategies for defeating the Runge Phenomenon for the approximation of non-periodic functions, part two: Multi-interval polynomial schemes and multidomain Chebyshev interpolation
Approximating a smooth function from its values f(x"i) at a set of evenly spaced points x"i through P-point polynomial interpolation often fails because of divergence near the endpoints, the ''Runge Phenomenon''. This report shows how to achieve an ...
A non-uniform basis order for the discontinuous Galerkin method of the acoustic and elastic wave equations
Here, we solve the time-dependent acoustic and elastic wave equations using the discontinuous Galerkin method for spatial discretization and the low-storage Runge-Kutta and Crank-Nicolson methods for time integration. The aim of the present paper is to ...
Strong-stability-preserving 3-stage Hermite-Birkhoff time-discretization methods
Strong-stability-preserving (SSP) time-discretization methods have a nonlinear stability property that makes them particularly suitable for the integration of hyperbolic conservation laws. A collection of SSP explicit 3-stage Hermite-Birkhoff methods of ...
Composition of stochastic B-series with applications to implicit Taylor methods
In this article, we construct a representation formula for stochastic B-series evaluated in a B-series. This formula is used to give for the first time the order conditions of implicit Taylor methods in terms of rooted trees. Finally, as an example we ...
Convergence of an adaptive Kačanov FEM for quasi-linear problems
We design an adaptive finite element method to approximate the solutions of quasi-linear elliptic problems. The algorithm is based on a Kacanov iteration and a mesh adaptation step is performed after each linear solve. The method is thus inexact because ...
A numerical method for mass conservative coupling between fluid flow and solute transport
We present a new coupled discretization approach for species transport in an incompressible fluid. The Navier-Stokes equations for the flow are discretized by the divergence-free Scott-Vogelius element on barycentrically refined meshes guaranteeing LBB ...
RKDG methods with WENO type limiters and conservative interfacial procedure for one-dimensional compressible multi-medium flow simulations
In this paper, we continue on studying the Runge-Kutta discontinuous Galerkin (RKDG) methods to solve compressible multi-medium flow with conservative treatment of the moving material interface. Comparing with the paper by J. Qiu, T.G. Liu and B.C. Khoo ...
Numerical approximation of a convolution model of ·θ-neuron networks
In this article, we consider a nonlinear integro-differential equation that arises in a @q@?-neural networks modeling. We analyze boundedness and invertibility of the model operator, construct approximate solutions using piecewise polynomials in space, ...
Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations
In this paper, alternating direction implicit compact finite difference schemes are devised for the numerical solution of two-dimensional Schrodinger equations. The convergence rates of the present schemes are of order O(h^4+@t^2). Numerical experiments ...
Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem
In this paper, we propose a method to improve the convergence rate of the lowest order Raviart-Thomas mixed finite element approximations for the second order elliptic eigenvalue problem. Here, we prove a supercloseness result for the eigenfunction ...
A non-standard symmetry-preserving method to compute bounded solutions of a generalized Newell-Whitehead-Segel equation
In this work, we propose a finite-difference scheme to approximate the solutions of a generalization of the classical, one-dimensional, Newell-Whitehead-Segel equation from fluid mechanics, which is an equation for which the existence of bounded ...