scholarly journals High-Order Asymptotic-Preserving Methods for Fully Nonlinear Relaxation Problems

2014 ◽  
Vol 36 (2) ◽  
pp. A377-A395 ◽  
Author(s):  
Sebastiano Boscarino ◽  
Philippe G. LeFloch ◽  
Giovanni Russo
Keyword(s):  
High Order ◽  
Asymptotic Preserving ◽  
Fully Nonlinear ◽  
Nonlinear Relaxation

High-Order Conservative Asymptotic-Preserving Schemes for Modeling Rarefied Gas Dynamical Flows with Boltzmann-BGK Equation

2015 ◽  
Vol 18 (4) ◽  
pp. 1012-1049 ◽  
Author(s):  
Manuel A. Diaz ◽  
Min-Hung Chen ◽  
Jaw-Yen Yang
Keyword(s):  
Rarefied Gas ◽  
Collision Operator ◽  
High Order ◽  
Gas Flows ◽  
Correction Procedure ◽  
Bgk Model ◽  
Time Step ◽  
Asymptotic Preserving ◽  
Bgk Equation ◽  
Rarefied Gas Flows

AbstractHigh-order and conservative phase space direct solvers that preserve the Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas flows are explored and studied. The approach is based on the conservative discrete ordinate method for velocity space by using Gauss Hermite or Simpsons quadrature rule and conservation of macroscopic properties are enforced on the BGK collision operator. High-order asymptotic-preserving time integration is adopted and the spatial evolution is performed by high-order schemes including a finite difference weighted essentially non-oscillatory method and correction procedure via reconstruction schemes. An artificial viscosity dissipative model is introduced into the Boltzmann-BGK equation when the correction procedure via reconstruction scheme is used. The effects of the discrete velocity conservative property and accuracy of high-order formulations of kinetic schemes based on BGK model methods are provided. Extensive comparative tests with one-dimensional and two-dimensional problems in rarefied gas flows have been carried out to validate and illustrate the schemes presented. Potentially advantageous schemes in terms of stable large time step allowed and higher-order of accuracy are suggested.

Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs

2015 ◽  
Vol 18 (5) ◽  
pp. 1482-1503 ◽  
Author(s):  
Tao Kong ◽  
Weidong Zhao ◽  
Tao Zhou
Keyword(s):  
Backward Stochastic Differential Equation ◽  
High Order ◽  
Cauchy Problems ◽  
Numerical Schemes ◽  
Parabolic Pdes ◽  
Hjb Equations ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Nonlinear Parabolic Pde

AbstractIn this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

Sign in / Sign up

Export Citation Format

Share Document