Numerical solutions of a single conservation equation by a fifth order WENO difference lattice Boltzmann method. (Chinese. English summary) Zbl 1289.65199
Summary: A fifth-order weighted essentially non-oscillatory (WENO)-lattice Boltzmann method for solving a one dimension conservation equation is developed. We define a generalized lattice Boltzmann distribution function, which can be calculated by the WENO difference schemes. Then the solving of the one dimension conservation equation becomes the solving of the generalized lattice Boltzmann distribution function by the WENO difference schemes. The method remains the benefits of the WENO method such as high precision and high resolution, and provides a new way to construct the lattice Boltzmann models for solving the Euler equation. This makes the lattice Boltzmann method more convenient to solve the compressible flow problems. As an example, we present 1D Burgers conservation equation, solve it by using the method we developed in this paper, and analyze the accuracy and stability of the model. At last, we make a summary of the method, and present problems to be further studied.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 76N15 | Gas dynamics (general theory) |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
