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A Hopf-Cole transformation based asymptotic method for kinetic equations with a Bhatnagar-Gross-Krook operator
| Content Provider | Semantic Scholar |
|---|---|
| Author | Luo, Songting Payne, Nicholas |
| Copyright Year | 2016 |
| Abstract | We present an effective asymptotic method for approximating the density of particles for kinetic equations with a Bhatnagar-Gross-Krook (BGK) relaxation operator in the large scale hyperbolic limit. The density of particles is transformed as the Hopf-Cole transformation, where the phase function is expanded as a power series with respect to the Knudsen number. The expansion terms can be determined with solutions of a sequence of equations. In particular, the leading order term is the viscosity solution of an effective HamiltonJacobi equation, and the higher order terms can be determined with the solutions of a sequence of transport equations. Both the effective Hamilton-Jacobi equation and the transport equations are formulated and solved in the physical space with necessary components determined as integrals in the velocity variable. Such integrals can be evaluated efficiently with Gauss quadrature rules. And the solutions of the equations do not depend on the Knudsen number. Therefore the expansion terms can be computed efficiently and used to approximate the phase function and the density function faithfully. In this work, the zeroth, first and second order terms in the expansion are used to obtain second order accuracy with respect to the Knudsen number. The proposed method balances efficiency and accuracy, and has the potential to deal with kinetic equations with more general BGK models. Numerical experiments verify the effectiveness of the proposed method. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://orion.math.iastate.edu/luos/Papers/KineticSimpleBGK.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
