A Direct Method for Modeling and Simulations of Elliptic and Parabolic Interface Problems

Interface problems have many applications in physics. In this dissertation, we develop a direct method for solving three-dimensional elliptic interface problems and study their application in solving parabolic interface problems. As many of the physical applications of interface problems can be approximated with partial differential equations (PDE) with piecewise constant coefficients, our derivation of the model is focused on interface problems with piecewise constant coefficients but have a finite jump across the interface. The critical characteristic of the method is that our computational framework is based on a finite difference scheme on a uniform Cartesian grid system and does not require an augmented variable as in the augmented approach. So the implementation of the method is easier to understand for non-experts in the area. The discretization of the PDE uses the standard seven-point central difference scheme for grid points away from the interface and a twenty-seven-point compact scheme that considers the jump discontinuities in the solution, flux, and jump ratio for grid points near or on the interface. As a result, the developed model can obtain second-order accuracy globally for both the solution and the solution\u27s gradient. Moreover, our numerical experiment indicates that eigenvalues of the coefficient matrix of the resulting linear system for the finite difference scheme are located in the left half-plane, implicating our method\u27s stability. We have also developed a model for solving two and three-dimensional parabolic interface problems using the Crank-Nicolson scheme and some modifications into the direct immersed interface method (IIM). The developed model can accurately capture the discontinuities in the solution across the interface and achieve second-order accuracy for both the solution and the solution\u27s gradient in both space and time

Matrix-equation-based strategies for convection-diffusion equations

We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on the matrix equation formulation of the problem are proposed, which naturally approximate the original discretized problem. For certain types of convection coefficients, we show that the explicit solution of the matrix equation can effectively replace the linear system solution. Numerical experiments with data stemming from two and three dimensional problems are reported, illustrating the potential of the proposed methodology

Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

Solving second-order linear ordinary differential equations by using interactive software

Differential equations constitute an area of great theoretical research and applications in several branches of science and technology. The scope of this work is to present new software that is able to show all the steps in the process of solving a linear second-order ordinary differential equation with constant coefficients.info:eu-repo/semantics/publishedVersio