# Differential quadrature solution of nonlinear reaction-diffusion equation with relaxation-type time integration

2009-01-01
Meral, G.
Tezer, Münevver
This paper presents the combined application of differential quadrature method (DQM) and finite-difference method (FDM) with a relaxation parameter to nonlinear reaction-diffusion equation in one and two dimensions. The polynomial-based DQM is employed to discretize the spatial partial derivatives by using Gauss-Chebyshev-Lobatto points. The resulting system of ordinary differential equations is solved, discretizating the time derivative by an explicit FDM. 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 stable solution. Numerical experiments are given to illustrate the scheme for one-dimensional Fisher-type problems and also for two-dimensional reaction-diffusion boundary-value problems. The agreement of the solution with the exact solution is very good in two-dimensional case while some other numerical schemes may result in some unwanted oscillations in the computed solution. Optimal value of the relaxation parameter is obtained numerically to prevent the use of very small time steps and to achieve stable solutions. The DQM with a relaxation-type finite-difference time integration scheme exhibits superior accuracy at large time values for the problems tending towards a steady state.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS

# Suggestions

 FORMULATION OF SOME GAUSSIAN INTEGRALS OVER R(N) VIA GENERATING-FUNCTIONS ERGENC, T; DEMIRALP, M (Informa UK Limited, 1994-01-01) This paper deals with the analytic formulation of the integrals over R(n) with the weight function exp(x(T)Cx) where the integrand is the product of quadratic forms x(T) A(j)x, j = 1,...,p, for arbitrary n x n symmetric matrices A(j). The technique is based on generating functions. First some functions are defined to generate these integrals for the special case A(j) = A(j) = A...A, and then practical formulas for the general case are derived.
 SOME INFINITE INTEGRALS INVOLVING PRODUCTS OF EXPONENTIAL AND BESSEL-FUNCTIONS Tezer, Münevver (Informa UK Limited, 1989-01-01) This paper is concerned with the evaluation of some infinite integrals involving products of exponential and Bessel functions. These integrals are transformed, through some identities, into the expressions containing modified Bessel functions. In this way, the difficulties associated with the computations of infinite integrals with oscillating integrands are eliminated.
 Least-squares differential quadrature time integration scheme in the dual reciprocity boundary element method solution of diffusive-convective problems Bozkaya, Canan (Elsevier BV, 2007-01-01) Least-squares differential quadrature method (DQM) is used for solving the ordinary differential equations in time, obtained from the application of dual reciprocity boundary element method (DRBEM) for the spatial partial derivatives in diffusive-convective type problems with variable coefficients. The DRBEM enables us to use the fundamental solution of Laplace equation, which is easy to implement computation ally. The terms except the Laplacian are considered as the nonhomogeneity in the equation, which ar...
 Solution to transient Navier-Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method Bozkaya, Canan; Tezer, Münevver (Wiley, 2009-01-20) The two-dimensional time-dependent Navier-Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equati...
 Finite element error analysis of a variational multiscale method for the Navier-Stokes equations Volker, John; Kaya Merdan, Songül (Springer Science and Business Media LLC, 2008-01-01) The paper presents finite element error estimates of a variational multiscale method (VMS) for the incompressible Navier-Stokes equations. The constants in these estimates do not depend on the Reynolds number but on a reduced Reynolds number or on the mesh size of a coarse mesh.
Citation Formats
G. Meral and M. Tezer, “Differential quadrature solution of nonlinear reaction-diffusion equation with relaxation-type time integration,” INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS, pp. 451–463, 2009, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/47187. 