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.