Non-uniform mesh high-order discretization for the generalized Burgers Equations

Using several properties of the polynomial splines, we first discuss the discretization of the first order spatial derivatives at different nodal points. Using such discretization we derive a scheme that is fourthorder accurate for the numerical solution of parabolic PDEs on a non-uniform mesh. Finally, we discuss the stability theory and compute the numerical results to illustrate the reliability of the scheme

A family of quasi-variable meshes high-resolution compact operator scheme for Burger's-Huxley, and Burger's-Fisher equation: Quasi-variable meshes compact operator scheme for Burger's type PDEs

We describe a quasi-variable meshes implicit compact finite-difference discretization having an accuracy of order four in the spatial direction and second-order in the temporal direction for obtaining numerical solution values of generalized Burger’s-Huxley and Burger’s-Fisher equations. The new difference scheme is derived for a general one-dimension quasi-linear parabolic partial differential equation on a quasi-variable meshes network to the extent that the magnitude of local truncation error of the high-order compact scheme remains unchanged in case of uniform meshes network. Practically, quasi-variable meshes high-order compact schemes yield more precise solution compared with uniform meshes high-order schemes of the same magnitude. A detailed exposition of the new scheme has been introduced and discussed the Fourier analysis based stability theory. The computational results with generalized Burger’s-Huxley equation and Burger’s-Fisher equation are obtained using quasi-variable meshes high-order compact scheme and compared with a numerical solution using uniform meshes high-order schemes to demonstrate capability and accuracy