Validated forward integration scheme for parabolic PDEs via Chebyshev series

Multirate Numerical Integration for Parabolic PDEs

To solve PDE problems with different time scales that are localized in space, multirate time integration is examined. This technique enables one to use large time steps for slowly time-varying spatial regions, and small steps for rapidly varying ones. Multirate time stepping is coupled with the local uniform grid refinement and provides a robust and efficient method for the target problem class...

A numerical scheme for solving nonlinear backward parabolic problems

‎In this paper a nonlinear backward parabolic problem in one‎ ‎dimensional space is considered‎. ‎Using a suitable iterative‎ ‎algorithm‎, ‎the problem is converted to a linear backward parabolic‎ ‎problem‎. ‎For the corresponding problem‎, ‎the backward finite‎ ‎differences method with suitable grid size is applied‎. ‎It is shown‎ ‎that if the coefficients satisfy some special conditions‎, ‎th...

Intermittency and multifractality: A case study via parabolic stochastic PDEs∗

Let ξ denote space-time white noise, and consider the following stochastic partial differential equations: (i) u̇ = 1 2 u′′ + uξ, started identically at one; and (ii) Ż = 1 2 Z′′ + ξ, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that th...

Validated Continuation for Equilibria of PDEs

One of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor-corrector continuation techniques. In the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output. We introduce a new technique ...

Infinite dimensional backstepping for nonlinear parabolic PDEs