Numerical solution of Burger’s equation via RBF-collocation scheme

A classical problem in computational hydraulics and/or hydrology is to quantify the spatial as well as temporal distribution of pressure and velocity fields within a water body. This quantification plays an important role in analysis and design of hydraulic structures, irrigation and drainage networks, and water transmission lines. In this regard, numerous numerical techniques have been devised to analyze the governing equations (e.g., Navier-Stokes equations) and its variants such as Burger’s equation with new scheme usually emerging from new technology. In this study, RBF-collocation scheme is implemented to solve the one dimensional nonlinear Burger’s equation. This particular equation is chosen to better assess and evaluate the performance of the cited numerical scheme as the equation possesses analytical exact solution under certain conditions. Conversion of partial differential equation into system of nonlinear algebraic equations is achieved by spatial approximation of the governing equation via RBF-collocation and temporal approximation via finite difference schemes. In contrast to the current research on RBF-collocation where Radial Basis Functions are considered to capture the spatial variation and RBF coefficients are introduced to mimic the temporal variation resulting in system of nonlinear ODEs in terms of coefficients, a special form of RBF-collocation is proposed to keep the original state variable(s) intact. The new proposal was found to be efficient and robust. Performance assessment is achieved by comparing the numerical solution with analytical solution available in the relevant literature. Results show that the RBF-collocation scheme is quite stable and accurate with two main advantages. First, the structures of RBF shape functions allow appropriate selection of nodal distribution consistent with process physics. Secondly, RBF-collocation is quite capable of solving the governing partial differential equations using a few grid points. In conclusion, RBF-collocation is considered to be an efficient scheme for obtaining numerical solution of Burger’s equation.

کلیدواژه ها:

RBF-collocation ، shape parameter ، Burger's equation