LARGE TIME STEP GENERALIZATION OF RANDOM CHOICE FINITE DIFFERENCE SCHEME FOR HYPERBOLIC CONSERVATION LAWS

A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of their conserved quantities. We show consistency of the scheme for arbitrarily large Courant numbers. For scalar problems the scheme is total variation diminishing.A brief discussion is given for entropy condition.

General Synthetic Iterative Scheme for Unsteady Rarefied Gas Flows

In rarefied gas flows,the spatial grid size could vary by several orders of magnitude in a single flow configuration(e.g.,inside the Knudsen layer it is at the order of mean free path of gas molecules,while in the bulk region it is at a much larger hydrodynamic scale).Therefore,efficient implicit numerical method is urgently needed for time-dependent problems.However,the integro-differential nature of gas kinetic equations poses a grand challenge,as the gain part of the collision operator is non-invertible.Hence an iterative solver is required in each time step,which usually takes a lot of iterations in the(near)continuum flow regime where the Knudsen number is small;worse still,the solution does not asymptotically preserve the fluid dynamic limit when the spatial cell size is not refined enough.Based on the general synthetic iteration scheme for steady-state solution of the Boltzmann equation,we propose two numerical schemes to push the multiscale simulation of unsteady rarefied gas flows to a new boundary,that is,the numerical solution not only converges within dozens of iterations in each time step,but also asymptotically preserves the Navier-Stokes-Fourier limit in the continuum flow regime,when the spatial grid is coarse,and the time step is large(e.g.,in simulating the extreme slow decay of two-dimensional Taylor vortex,the time step is even at the order of vortex decay time).The properties of fast convergence and asymptotic preserving of the proposed schemes are not only rigorously proven by the Fourier stability analysis for simplified gas kinetic models,but also demonstrated by several numerical examples for the gas kinetic models and the Boltzmann equation.

Fast-Converging and Asymptotic-Preserving Simulation of Frequency Domain Thermoreflectance

The heat conduction under fast external excitation exists in many experiments measuring the thermal conductivity in solids,which is described by the phonon Boltzmann equation,i.e.,the Callaway’s model with dual relaxation times.Such a kinetic system has two spatial Knudsen numbers related to the resistive and normal scatterings,and one temporal Knudsen number determined by the external oscillation frequency.Thus,it is a challenge to develop an efficient numerical method.Here we first propose the general synthetic iterative scheme(GSIS)to solve the phonon Boltzmann equation,with the fast-converging and asymptotic-preserving properties:(i)the solution can be found within dozens of iterations for a wide range of Knudsen numbers and frequencies,and(ii)the solution is accurate when the spatial cell size in the bulk region is much larger than the phonon mean free path.Then,we investigate how the heating frequency affects the heat conduction in different transport regimes.