A flexible solution method for the initial-boundary value problem of the temperature field in a one-dimensional domain of a solid with significantly nonlinear material parameters and radiation boundary conditions is proposed. A transformation of the temperature values allows to isolate the nonlinear material characteristics into a single coefficient of the heat conduction equation. The Galerkin...

The objective is to construct and discretize (to fourth-order accuracy) a hybrid model for waveguide problems, where a finite difference method for inhomogeneous layers is coupled to a boundary element method (BEM) for a set of homogeneous layers. Such a hybrid model is adequate for underwater acoustics in complicated environments. Waveguides with either plane or axial symmetry are treated, whi...

In this paper, a solution-adaptive algorithm is presented for the simulation of incompressible viscous flows. The framework of this method consists of an adaptive local stencil refinement algorithm and 3-points central difference discretization. The adaptive local stencil refinement is designed in such a manner that 5-points symmetric stencil is guaranteed at each interior node, so that convent...

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter 0 < ε ≪ 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagat...

We consider the two-dimensional, time-dependent Schrödinger equation discretized with the Crank-Nicolson finite difference scheme. For this difference equation we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. We apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron in...

We analyze a Crank–Nicolson–type finite difference scheme for the Kuramoto– Sivashinsky equation in one space dimension with periodic boundary conditions. We discuss linearizations of the scheme and derive second–order error estimates.

Numerical computation of textile permeability is important for composite manufacturing. Using Darcy’s law, permeability can be derived from a simulation of the fluid flow, i.e. after solving the Stokes, Navier-Stokes or Brinkman equations. The latter allow to model intra-yarn flow in case of permeable yarns. In this paper we present a numerical method for the calculation of the permeability of ...

In this paper, we give a fourth-order compact finite difference scheme for the general forms of two point boundary value problems and two dimensional elliptic partial differential equations (PDE’s). By decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problem, we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix. Thus...

We develop systematically a numerical approximation strategy to discretize a hydrodynamic phase field model for a binary fluid mixture of two immiscible viscous fluids, derived using the generalized Onsager principle that warrants not only the variational structure but also the energy dissipation property. We first discretize the governing equations in space to arrive at a semi-discretized, tim...

In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and...