A time-space B-spline integrator for the Burgers' equation

The purpose of this paper is to propose a new algorithm for obtaining approximate solutions to the Burgers' equation (BE). Integration in time by a quadratic B-spline collocation method is shown. To the best of our knowledge, B-splines have not previously been used to integrate partial differential equations in both time and space. First, the BE is integrated using quadratic B-spline functions in time, and then the time-integrated BE is further solved in space via the cubic B-spline collocation method. The resulting recursive algebraic equation is used to obtain both shock wave and front propagation solutions of the BE, demonstrating the effectiveness of the space--time B-spline collocation method.