A highly accurate algorithm for retrieving the predicted behavior of problems with piecewise-smooth initial data

Abstract

A numerical scheme is constructed for the second-order parabolic partial differential equation with piecewise smooth initial data. The scheme comprises an orthogonal spline collocation strategy with the Rannacher time-marching. The irregular behavior of the underlying initial conditions of such differential equations results in inaccurate approximations due to the quantization error. For such problems, even the A-stable Crank-Nicolson scheme yields only first-order convergence in the temporal direction, with oscillations near the discontinuity. Applying mathematical perspective to dampen these oscillations, we present a highly accurate orthogonal spline collocation method with a smooth but straightforward time-marching scheme that significantly improves the convergence order. Through rigorous analysis, the present conjunctive scheme's convergence in the spatial direction is shown fourth-order (in and -norms) and third-order (in -norm), and it is shown second-order in the temporal direction. The performance and robustness of the contributed scheme are conclusively demonstrated with two test examples.