Numerical Blowup Solutions for Boundary Value Models

Abstract

In this thesis, we discuss several numerical methods to approximate singular solutions for some partial differential equations such as Burgers’ equation, Prandtl’s equations, and the inviscid primitive equations. The numerical solutions we obtain for Burgers’ equation and Prandtl’s equations are compared with the existing analytical and numerical solutions in the literature. We observe the singularity formation in the numerical solutions to Burgers’ equation and Prandtl’s equations in finite time. For the inviscid primitive equations with the initial data are close to a suitable rescale of a smooth blowup profile proven by Collot, Ibrahim, and Lin in [7], we compare the numerical solution to the theoretical blowup profile. The solution we obtain from the numerical scheme follows the profile, but the difference between the numerical and analytical profiles is quite significant closer to the blowup time. We then examine the stability of the numerical solutions by considering a small perturbation for the initial data. The gap between the perturbed and unperturbed solutions reduces as we choose smaller perturbation. However, this gap grows as it approaches the blowup time, and the stability of the numerical solutions remains in doubt.