Abstract
This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver.
Highlights
Many systems of interest in biology and chemistry have successfully been modelled by partial differential equations (PDEs) exhibiting an oscillatory or periodic solution
The specific path we aim to follow in order to numerically solve a diffusion PDE by exponentially fitted methods is essentially the following: we first introduce and analyze exponentially fitted numerical differentiation formulae which approximate the second derivative ∂2u/∂x2, as described in Section “An exponentially fitted second order finite difference”; we consider diffusion PDEs with mixed boundary conditions and provide a spatial semi-discretization of the problem in Section “A test problem: diffusion equation with mixed boundary conditions”; we provide numerical experiments in Section “Numerical results on the semi-discrete model”, where the proposed approach is tested and compared with others known from the existing literature
We have presented an alternative approach to numerically solve partial diffential equations. This approach is based on the exponential fitting technique, which consists in specializing a numerical method to the behaviour of the solution
Summary
Many systems of interest in biology and chemistry have successfully been modelled by partial differential equations (PDEs) exhibiting an oscillatory or periodic solution. We mention oscillatory reactiondiffusion equations, which have periodic waves as fundamental solutions. Usually denoted in the literature as diffusion equation (compare, for instance, (Hamdi et al 2007; Isaacson and Keller 1994) and references therein). Such an equation is called Fourier Second Law when applied to heat transfer; in this case, the function u(x, t) represents the temperature (evolving both in space and in time), while the constant δ is the thermal diffusivity of the material. Eq 1 is employed, for instance, to model mass diffusion: in this case, it is better known as Fick Second Law, u(x, t) represents the mass concentration and δ is the mass
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