Invited Lecture
Implicit-explicit schemes for PDE with convection and degenerate diffusion
Universitat de València, Spain
Abstract
When using the method of lines for PDE with convection and (possibly strong)
degenerate diffusion,
Implicit-Explicit (IMEX) Runge-Kutta (RK) methods,
that combine an explicit RK scheme for the time integration of the
convective part with a diagonally implicit one for the diffusive
part, are suitable for the much more favourable stability
restrictions with respect to explicit integrators and for not having
to deal, as fully implicit solvers do, with the fairly sophisticated
discretization of the convective terms when they are dominant.
In [Bürger, Mulet, Villada, SISC, 2013] a scheme of this type is
proposed, for which the nonlinear and nonsmooth systems of algebraic
equations arising in the implicit treatment of the degenerate
diffusive part are solved by smoothing of the diffusion coefficients
combined with a damped Newton-Raphson method with a line search
strategy for globalizing convergence.
To overcome the CPU and implementations costs of these schemes while
keeping the advantageous stability properties of
IMEX-RK methods, a second variant of these methods is proposed in
[Boscarino et al., SISC, 2015], in which the diffusion terms are
discretized in a way that more carefully
distinguishes between stiff and nonstiff dependence, such that in each
Runge-Kutta stage only a linear system needs to be solved, still maintaining
high order accuracy in time.
These schemes may be advantageous
in some cases, but are not advisable in those cases where special
structure of the diffusive terms would be lost. This is the case of
some nonlinear convection-diffusion equations with nonlocal flux
and possibly degenerate diffusion that arise in many scientific
contexts [Carrillo, Chertock, Huang, CCP, 2015],
[Bürger, Inzunza, Mulet, NMPDE, 2019].
In this talk a survey of these techniques will be given, some recent
successful applications of them will be reported and some future
applications, as multispecies nonlinear nonlocal equations
with cross-diffusion or Navier-Stokes-Cahn-Hilliard equations,
will be presented.
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