LINEARLY IMPLICIT IMEX RUNGE-KUTTA METHODS FOR A CLASS OF DEGENERATE CONVECTION-DIFFUSION PROBLEMS

摘要

Multispecies kinematic flow models with strongly degenerate diffusive corrections give rise to systems of nonlinear convection-diffusion equations of arbitrary size. Applications of these systems include models of polydisperse sedimentation and multiclass traffic flow. Implicit-explicit (IMEX) Runge-Kutta (RK) methods are suitable for the solution of these convection-diffusion problems since the stability restrictions, coming from the explicitly treated convective part, are much less severe than those that would be deduced from an explicit treatment of the diffusive term. These schemes usually combine an explicit RK scheme for the time integration of the convective part with a diagonally implicit one for the diffusive part. In [R. Burger, P. Mulet, and L. M. Villada, SIAM J. Sci. Comput., 35 (2013), pp. B751-B777] a scheme of this type is proposed, where 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 Newton-Raphson method with line search. This nonlinearly implicit method is robust but associated with considerable effort of implementation and possibly CPU time. To overcome these shortcomings while keeping the advantageous stability properties of IMEX-RK methods, a second variant of these methods is proposed in which the diffusion terms are discretized in a way that more carefully distinguishes between stiff and nonstiff dependence, such that in each time step only a linear system needs to be solved still maintaining high order accuracy in time, which makes these methods much simpler to implement. In a series of examples of polydisperse sedimentation and multiclass traffic flow, it is demonstrated that these new linearly implicit IMEX-RK schemes approximate the same solutions as the nonlinearly implicit versions, and in many cases these schemes are more efficient.