A time-space B-spline integrator for the Burgers' equation
Idris Dag, Serkan Uğurluoğlu, Nihat Adar
TL;DR
The Burgers' equation, $u_t + u u_x - \upsilon u_{xx} = 0$, presents challenges for small viscosity due to shock formation. The authors introduce a time-space B-spline collocation method that discretizes time with quadratic B-splines and space with cubic B-splines, yielding a space-time expansion $U(x,t)=\sum_j \delta^j(x) B^j(t)$ and $\delta(x,t^j)=\sum_i \sigma_i^j B_i(x)$, which leads to $n+1$ algebraic equations in $n+3$ unknowns and is solved via linearization and a few iterations. They validate the approach on two BE scenarios—a shock-like solution and a traveling-front solution—reporting $L_\infty$ and $L_2$ norms that demonstrate accuracy and robustness, with sharper fronts for smaller $\upsilon$ and localized higher errors near fronts. Overall, the space-time B-spline collocation framework offers a viable, alternative route to traditional time integrators for PDEs, and sets the stage for higher-order B-spline extensions to improve accuracy.
Abstract
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.