An adaptive tailored finite point method for the generalized Burgers' equations

A tailored finite point method, using a minimal machinery algorithm yet utilizing the initial conditions and local properties of the solution to the hilt, is proposed to serve as a global platform to solve the generalized Burgers' equation. On an explicit centred 4-point stencil, the nodal solutions at the advanced time level are written as a linear combination of the nodal solutions at the preceding level. The scalars in this linear combination are determined using a set of basis functions. The extraction of these basis functions is through the fundamental solutions derived via the method of separation of variables. This, in turn, brings in the influence of the local properties of the general solution. The nodal maximum of the initial conditions is chosen as the separation constant to actuate continuous dependence on the initial conditions. Withal this separation constant works well for the generalized Burgers' equation, thereby instituting a common platform to solve these class of equations. The non-linearity is taken care of through an iterative technique where the non-linear term is replaced by an anterior temporal level iterated value. Conditional stability is established through the von-Neumann stability analysis. The method is consistent with second-order convergence in spatial variables and first-order convergence in the temporal variable. Numerical experiments are conducted on multifarious examples, and the obtained results are very much in accordance with the available exact solutions. The numerical results of examples with no useable closed-form representation of the exact solution are vindicated through the double meshing principle. Also, the error analysis establishes that the method works pretty well on coarse meshes, wherefore cutting the computational cost and increasing rapidity. Despite being a simple and straightforward algorithm with no usage of elite techniques, the method stands on par with quite a few methods in the literature.

Automated summary

This paper proposes a tailored finite point method to solve the generalized Burgers' equation by utilizing initial conditions and local properties of the solution. The method shows stability and convergence, and achieves good results in numerical experiments.