Solution Of The Nonlinear Diffusion Equation
Using The Dual Reciprocity Boundary Element
Method And The Relaxation Type Time
Integration Scheme
Author(s)
G. Meral
Abstract
We present the combined application of the dual reciprocity boundary element
method (DRBEM) and the finite difference method (FDM) with a relaxation
parameter to the nonlinear diffusion equation: ∂u
∂t = ν∇2u + p(u) where p(u)
is the nonlinear term. The DRBEM is employed to discretize the spatial partial
derivatives by using the fundamental solution of the Laplace operator, keeping the
time derivative and the nonlinearity as the nonhomogeneous terms in the equation.
The resulting system of ordinary differential equations is solved using the
FDM with a relaxation procedure. A relaxation parameter is used to position the
solution from the two time levels aiming to increase the convergence rate with a
moderate time step to the steady state and also to obtain a stable solution. The nonlinear
terms do not present problems since they are treated as the nonhomogeneity
of the diffusion equation with the help of the DRBEM. The solution procedure
described here is also applicable to the diffusion and convection-diffusion equation
and can be considered as the extension to the nonlinear reaction-diffusion
equation. Numerical experiments are given to illustrate this scheme and to compare
its performance with the other numerical schemes as well as the exact solution
whenever it is available. The solution agrees very well with the exact solution
while some other numerical schemes may result with some unwanted oscillations
in the computed solution. The optimal value of the relaxation parameter is obtained
numerically for preventing use of very small time increments and to achieve a stable
solution. The DRBEM with a relaxation type time integration scheme exhibits
a superior accuracy at large time values for the problems tending towards a steady
state.
Keywords: dual reciprocity boundary element method, reaction-diffusion
equation.
Keywords
dual reciprocity boundary element method, reaction-diffusion
equation
Related Book
Other papers in this volume
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