Solving Parabolic Singularly Perturbed Problems by Collocation Using Tension Splines

Abstract

Tension spline is a function that, for given partition x₀ < x₁ < … < x_n, on each interval [x_i, x_i+1] satisfies differential equation (D⁴ − ρ ² _i D²)u = 0, where ρ_i's are prescribed nonnegative real numbers. In the literature, tension splines are used in collocation methods applied to two-points singularly perturbed boundary value problems with Dirichlet boundary conditions.

In this paper, we adapt collocation method to solve a time dependent reaction-diffusion problem of the form

$$\varepsilon ^2 \frac{{\partial ^2 u}} {{\partial x^2 _{} }} - c(x,t)u - p(x,t)\frac{{\partial u}} {{\partial t}} = f(x,t)$$

with Dirichlet boundary conditions. We tested our method on the time-uniform mesh with N_x × N_t elements. Numerical results show ε-uniformly convergence of the method.