Quintic B-Spline Galerkin Method for Numerical Solutions of the Burgers' Equation

The nonlinear Burgers' equation is solved numerically by a method of Galerkin using quintic B-splines as both shape and weight functions over the fi- nite intervals. The same method is applied to the time-split Burgers' equation. Numerical comparison of results of both algorithms and some other published numerical results is done by studying two standard problems. The nonlinear Burgers' equation has been used as test example for the numeri- cal methods since this equation can be solved analytically for the various boundary and initial conditions. So numerical results can be compared with analytical results. Nowadays many of the numerical methods have been engaged to get the solution of the Burgers' equation with small viscosity. With these smaller constants, numerical results are likely to produce the results having non-physical oscillations unless the sizes of both space and time steps are unrealistically small. Some of the methods such as collocation method and Petrov-Galerkin finite elements have been used to obtain accurate numerical solutions for small viscosity coefficients. Spline functions, which are a class of piecewise polynomials having continuity properties of up to the degree lower than that of the spline functions play an important role of setting ap- proximate functions. A type of splines known as B-splines are very much in use with Galerkin method to have functional approximation of the unknowns in differential equations. It provides the manageable band matrix system. This method previ- ously has been implemented to get numerical solution of Burgers' equation. The finite element method for solutions of the Burgers' equation based on a collocation method using cubic splines as interpolation functions is set up by L. R. T. Gardner et al. (2). The same method with the cubic B-splines instead of quadratic B-splines