Quasi-linear PDEs and low-dimensional sets

Solutions to Problems for Quasi-Linear PDEs

(a) At what time ts and position xs does a shock first form? (b) Identify the important values of s and find the corresponding characteristics. Sketch the characteristics and indicate the region in the xt-plane in which the solution is well-defined (i.e. does not break down). (c) Construct tables as in class for the x and u values at the important values of s for times t = 0, 2/3, 4/3. Use thes...

Projection Multilevel Methods for Quasi-linear PDEs: V-cycle Theory

The projection multilevel method can be an efficient solver for systems of nonlinear partial differential equations that, for certain classes of nonlinearities (including least-squares formulations of the Navier-Stokes equations), requires no linearization anywhere in the algorithm. This paper provides an abstract framework and establishes optimal V-cycle convergence theory for this method.

An interpolated stochastic algorithm for quasi-linear PDEs

In this paper, we improve the forward-backward algorithm for quasi-linear PDEs introduced in Delarue and Menozzi [8]. The new discretization scheme takes advantage of the standing regularity properties of the true solution through an interpolation procedure. For the convergence analysis, we also exploit the optimality of the square Gaussian quantization used to approximate the conditional expec...

A Forward-Backward Stochastic Algorithm for Quasi-Linear PDEs

We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled Forward-Backward SDEs, which provides an efficient probabilistic representation of this type of equations. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for t...

The method of characteristics applied to quasi-linear PDEs 18.303 Linear Partial Differential Equations