An implicit-explicit discontinuous Galerkin scheme using a newton-free Picard iteration for a thin-film model

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2019-01-01
Authors
Logemann, Caleb
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James . Rossmanith
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Mathematics
Abstract

This paper provides a high-order numerical scheme for solving thin-film models of the

form q_t + (q^2 - q^3)_x = -(q^3 q_{xxx})_x.

The second term in this equation is of nonlinear hyperbolic type, while the right-hand

side is of nonlinear parabolic type.

The nonlinear hyperbolic term is discretized with the standard modal discontinuous

Galerkin method, and the nonlinear parabolic term is discretized with the local

discontinuous Galerkin method.

Propagation in time is done with an implicit-explicit Runge-Kutta scheme so as to

allow for larger time steps.

The timestep restriction for these methods is determined by the hyperbolic wavespeed

restriction, and is not limited by the nonlinear parabolic term.

A novel aspect of this method is that the resulting nonlinear algebraic equations

are solved with a Newton-free iteration with a Picard iteration.

The number of iterations required to converge is less than or equal to the estimated

order of the method.

We have demonstrated with this method up to third order convergence.

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Sun Dec 01 00:00:00 UTC 2019