High order positivity-preserving discontinuous
Galerkin methods for steady state or implicit
time discretization of linear hyperbolic equations
Chi-Wang Shu
Division of Applied Mathematics
Brown University
High order discontinuous Galerkin (DG) methods are widely
used in solving linear and nonlinear hyperbolic equations.
One important property of such equations is positivity
of their solutions, in the sense that if the data (initial
condition, boundary condition, source terms) are positive
then the solution stays positive. It is a challenge to
maintain such positivity numerically for high order schemes.
In the past few years, significant progress has been made
to design positivity-preserving (PP) DG methods which
maintain high order accuracy with explicit strong
stability preserving time discretization. However, it is
more challenging to design implicit time discretization
or steady state DG solvers which have the same property.
In this talk we will describe our recent research on
designing high order PP DG methods for steady state or
implicit time discretization of linear hyperbolic
equations. (1) For time dependent problems with periodic
boundary conditions, we design high order in space PP DG
solver with backward Euler time discretization which can
maintain high order spatial accuracy with a lower bound
on the CFL number. This is a joint work with Tong Qin.
(2) For steady state or time dependent problems with
inflow boundary conditions, we design PP DG methods
which can maintain high order accuracy. In two spatial
dimensions this would involve an augmented DG space.
This is a joint work with Dan Ling and Juan Cheng.