scholarly journals Exact difference schemes for a two-dimensional convection–diffusion–reaction equation

2014 ◽  
Vol 67 (12) ◽  
pp. 2205-2217 ◽  
Author(s):  
Magdalena Lapinska-Chrzczonowicz ◽  
Piotr Matus
Keyword(s):  
Difference Schemes ◽  
Diffusion Reaction ◽  
Two Dimensional ◽  
Reaction Equation ◽  
Convection Diffusion ◽  
Exact Difference

scholarly journals Two-dimensional meshless solution of the non-linear convection diffusion reaction equation by the Local Hermitian Interpolation method

2013 ◽  
Vol 9 (17) ◽  
pp. 21-51 ◽  
Author(s):  
Carlos Bustamante ◽  
Henry Power ◽  
Whady Florez ◽  
Alan Hill Betancourt
Keyword(s):  
Numerical Scheme ◽  
Diffusion Reaction ◽  
Interpolation Function ◽  
Two Dimensional ◽  
Reaction Equation ◽  
Convection Diffusion ◽  
Non Linear ◽  
Newton Raphson ◽  
Hermitian Interpolation ◽  
Raphson Method

A meshless numerical scheme is developed for solving a generic version of the non-linear convection-diffusion-reaction equation in two-dimensional do-mains. The Local Hermitian Interpolation (LHI) method is employed for thespatial discretization and several strategies are implemented for the solution of the resulting non-linear equation system, among them the Picard iteration, the Newton Raphson method and a truncated version of the Homotopy Analysis Method (HAM). The LHI method is a local collocation strategy in which Radial Basis Functions (RBFs) are employed to build the interpolation function. Unlike the original Kansa’s Method, the LHI is applied locally and the boundary and governing equation differential operators are used to obtain the interpolation function, giving a symmetric and non-singular collocation matrix. Analytical and Numerical Jacobian matrices are tested for the Newton-Raphson method and the derivatives of the governing equation with respect to the homotopy parameter are obtained analytically. The numerical scheme is verified by comparing the obtained results to the one-dimensional Burgers’ and two-dimensional Richards’ analytical solutions. The same resultsare obtained for all the non-linear solvers tested, but better convergence ratesare attained with the Newton Raphson method in a double iteration scheme.

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