62 research outputs found
High Order Asymptotic Preserving DG-IMEX Schemes for Discrete-Velocity Kinetic Equations in a Diffusive Scaling
In this paper, we develop a family of high order asymptotic preserving
schemes for some discrete-velocity kinetic equations under a diffusive scaling,
that in the asymptotic limit lead to macroscopic models such as the heat
equation, the porous media equation, the advection-diffusion equation, and the
viscous Burgers equation. Our approach is based on the micro-macro
reformulation of the kinetic equation which involves a natural decomposition of
the equation to the equilibrium and non-equilibrium parts. To achieve high
order accuracy and uniform stability as well as to capture the correct
asymptotic limit, two new ingredients are employed in the proposed methods:
discontinuous Galerkin spatial discretization of arbitrary order of accuracy
with suitable numerical fluxes; high order globally stiffly accurate
implicit-explicit Runge-Kutta scheme in time equipped with a properly chosen
implicit-explicit strategy. Formal asymptotic analysis shows that the proposed
scheme in the limit of epsilon -> 0 is an explicit, consistent and high order
discretization for the limiting equation. Numerical results are presented to
demonstrate the stability and high order accuracy of the proposed schemes
together with their performance in the limit
Asymptotic preserving and uniformly unconditionally stable finite difference schemes for kinetic transport equations
In this paper, uniformly unconditionally stable first and second order finite
difference schemes are developed for kinetic transport equations in the
diffusive scaling. We first derive an approximate evolution equation for the
macroscopic density, from the formal solution of the distribution function,
which is then discretized by following characteristics for the transport part
with a backward finite difference semi-Lagrangian approach, while the diffusive
part is discretized implicitly. After the macroscopic density is available, the
distribution function can be efficiently solved even with a fully implicit time
discretization, since all discrete velocities are decoupled, resulting in a
low-dimensional linear system from spatial discretizations at each discrete
velocity. Both first and second order discretizations in space and in time are
considered. The resulting schemes can be shown to be asymptotic preserving (AP)
in the diffusive limit. Uniformly unconditional stabilities are verified from a
Fourier analysis based on eigenvalues of corresponding amplification matrices.
Numerical experiments, including high dimensional problems, have demonstrated
the corresponding orders of accuracy both in space and in time, uniform
stability, AP property, and good performances of our proposed approach
High order asymptotic preserving scheme for linear kinetic equations with diffusive scaling
In this work, high order asymptotic preserving schemes are constructed and
analysed for kinetic equations under a diffusive scaling. The framework enables
to consider different cases: the diffusion equation, the advection-diffusion
equation and the presence of inflow boundary conditions. Starting from the
micro-macro reformulation of the original kinetic equation, high order time
integrators are introduced. This class of numerical schemes enjoys the
Asymptotic Preserving (AP) property for arbitrary initial data and degenerates
when goes to zero into a high order scheme which is implicit for the
diffusion term, which makes it free from the usual diffusion stability
condition. The space discretization is also discussed and high order methods
are also proposed based on classical finite differences schemes. The Asymptotic
Preserving property is analysed and numerical results are presented to
illustrate the properties of the proposed schemes in different regimes
Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)
This workshop brought together
leading experts, as well as the most
promising young researchers, working on nonlinear
hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in
modeling, analysis, and numerics. Particular topics included ill-/well-posedness,
randomness and multiscale modeling, flows in a moving domain, free boundary problems,
games and control
- …