First Order Multistep Method for Solving Diffusion Convection Equations Derivable from Polynomials as Basis Functions and its Applications

Abstract

In this work, a new numerical finite difference scheme for the solution of heat
diffusion conduction equation arising from heat conduction is developed due
to the recent growing interest in literatures in the derivation of continuous
numerical finite difference method for solving heat diffusion convection
equations. This was done based on the collocation and interpolation of the
heat diffusion convection equations directly over multi steps along lines but
without reduction to a system of Ordinary Differential Equations (ODE). The
intention was to avoid the cost of solving a large system of coupled ODEs often
arising from the reduction method by a semi - discretization. The
performance of the new numerical finite difference scheme was tested. The
numerical results obtained showed that the method provided the same results
with the known explicit finite difference method. There was no semidiscretization
involved in the derivation of this scheme, and no reduction of
the heat diffusion convection equations to a system of ODE is recorded, but
rather a system of algebraic equations is formulated. Therefore, the desire is
to derive a new numerical scheme that will be used in finding the solutions of
the system of algebraic equations formulated from the discretization of the
heat diffusion convection equations with respect to the space and time
variables. This new numerical method was applied to solve two different test
problems with known explicit solutions by Schmidt.