An improved method for solving quasilinear convection diffusion problems on a coarse mesh

TL;DR

This paper introduces a novel iterative method for solving quasilinear convection diffusion problems on coarse meshes, enhancing stability and convergence through adaptive regularization and mesh refinement strategies.

Contribution

It presents a new Newton-like iterative solver based on regularized pseudo-transient continuation with a modified Newmark integrator for improved stability on coarse, rough data meshes.

Findings

The method achieves q-linear local convergence.

Numerical examples confirm the predicted convergence rate.

The approach effectively captures boundary and internal layers.

Abstract

A method is developed for solving quasilinear convection diffusion problems starting on a coarse mesh where the data and solution-dependent coefficients are unresolved, the problem is unstable and approximation properties do not hold. The Newton-like iterations of the solver are based on the framework of regularized pseudo-transient continuation where the proposed time integrator is a variation on the Newmark strategy, designed to introduce controllable numerical dissipation and to reduce the fluctuation between the iterates in the coarse mesh regime where the data is rough and the linearized problems are badly conditioned and possibly indefinite. An algorithm and updated marking strategy is presented to produce a stable sequence of iterates as boundary and internal layers in the data are captured by adaptive mesh partitioning. The method is suitable for use in an adaptive framework making use of local error indicators to determine mesh refinement and targeted regularization. Derivation and q-linear local convergence of the method is established, and numerical examples demonstrate the theory including the predicted rate of convergence of the iterations.