WITHDRAWN: A Space-Time Collocation Method for Solving Burger's Equation via Convex Relaxation

The ability to solve partial differential equations accurately, robustly, and scalably is a core aspect of computational fluid dynamics and, by extension, computational aerospace shape optimization. In this paper, we investigate a novel approach to the solution of nonlinear PDEs in conservation form. In particular, we consider a spectral approach where the complete space--time solution is approximated as a polynomial-augmented network of radial basis functions. Our approach extends Kansa's method to nonlinear PDEs in conservation form by introducing a convex relaxation for the flux term. Although the approach is quite general, we demonstrate its efficacy in solving the one-dimensional viscous Burger's equation. In addition to being meshless, this approach allows future exploration of one-shot convex optimization approaches to aerodynamic shape optimization.