Finite difference scheme for linear advection equation with dependence on scalar dimensionless parameter is constructed. Stability of the scheme is investigated by von Neumann method and stability domain in parameter space is constructed. Dispersive and dissipative properties of the scheme are optimized by the choice of scalar parameter. Low dispersion and dissipation of the scheme is demonstrated by the numerical solution of simple test Cauchy problem. Scheme constructed may be applied in computations based on splitting method for Boltzmann or lattice Boltzmann equations.
The approach to optimization of finite-difference (FD) schemes for the linear advection equation (LAE) is proposed. The FD schemes dependent on the scalar dimensionless parameter are considered. The parameter is included in the expression, which approximates the term with spatial derivatives. The approach is based on the considering of the dispersive and dissipative characteristics of the schemes as the functions of the parameter. For the proper choice of the parameter, these functions are minimized. The approach is applied to the optimization of two-step schemes with an asymmetric approximation of time derivative and with various approximations of the spatial term. The cases of schemes from first to fourth approximation orders are considered. The optimal values of the parameter are obtained. Schemes with the optimal values are applied to the solution of test problems with smooth and discontinuous initial conditions. Also, schemes are used in the FD-based lattice Boltzmann method (LBM) for modeling of the compressible gas flow. The obtained numerical results demonstrate the convergence of the schemes and decaying of the numerical dispersion.
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