Solution Of The Nonlinear Diffusion Equation Using The Dual Reciprocity Boundary Element Method And The Relaxation Type Time Integration Scheme
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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.
dual reciprocity boundary element method, reaction-diffusion equation