Yin-Tzer Shih

A Tailored Finite Point Method for Convection-Diffusion-Reaction Problems

We study a tailored finite point method (TFPM) for solving the convection-diffusion-reaction equation. The solution basis functions for the TFPM are constructed for a 5 point, 7 point and 9 point stencil. Some truncation error calculations are given. Numerical tests are given on problems containing a boundary or interior layer. The tests compare TFPM with several versions of a Petrov-Galerkin finite element schemes, and suggest that TFPM gives a superior resolution of the layers.

Characteristic Tailored Finite Point Method for Convection-Dominated Convection-Diffusion-Reaction Problems

In this paper, we propose a characteristic tailored finite point method (CTFPM) for solving the convection-diffusion-reaction equation with variable coefficients. We develop an algorithm to construct a streamline-aligned grid for the CTFPM. Our numerical tests show for small diffusion coefficient the CTFPM solution resolves the internal and boundary layers regardless the mesh size, and depicts that CTFPM method with a streamline grid has excellent performance compared with the tailored finite point method and a streamline upwind finite element method when ε is small.

Tailored Finite Point Methods for Solving Singularly Perturbed Eigenvalue Problems with Higher Eigenvalues

We study tailored finite point methods (TFPM) for solving the singularly perturbed eigenvalue (SPE) problems. We first provide an asymptotic analysis for the eigenpairs and show that for some special potential functions when

A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with Boundary Layers

\varepsilon

A Novel Technique for Constructing Difference Schemes for Systems of Singularly Perturbed Equations

ε approaches to zero the square of eigenfunction converges to a Dirac delta function weakly, and the eigenvalue converges to the minimum value of the potential function. For computing the eigenfunction with higher eigenvalue we propose two variants of TFPM for one-dimensional SPE problems and a nonlinear least square TFPM for two-dimensional problems. The eigenfunction with higher eigenvalue can be easily computed on a related coarse mesh on numerical tests, and suggests that the proposed schemes are accurate and efficient for the SPE problems.

Tailored Finite Point Method for Numerical Solutions of Singular Perturbed Eigenvalue Problems

In this paper we propose an explicit and implicit tailored finite point (EITFP) method for solving a finance object—the European option pricing. We derive a diffusion equation from the Black–Scholes equation in dealing with both European call option and European put option. The performance of the EITFP has been compared with popular numerical schemes, and the numerical experiment shows that the EITFP is accurate. Furthermore, the EITFP is efficient for being implemented by using a multi-core parallelized acceleration with CPU and Graphics Processing Unit (GPU) for the option computation.