Изучены монотонность и точность модифицированной схемы КАБАРЕ, аппроксимирующей квазилинейную гиперболическую систему законов сохранения. Получены условия, при которых эта схема сохраняет монотонность разностного решения относительно инвариантов линейного приближения аппроксимируемой системы. В качестве конкретного примера рассмотрена аппроксимация системы законов сохранения теории мелкой воды. На примере этой системы показано, что подобно TVD-схемам повышенного порядка аппроксимации на гладких решениях схема КАБАРЕ, несмотря на высокую точность при локализации ударных волн, снижает свой порядок сходимости в областях их влияния.
We studied the monotonicity and the accuracy of the modified CABARET scheme approximating the quasilinear hyperbolic system of conservation laws. We obtained conditions under which this scheme preserves the monotonicity of the finite-difference solution with respect to invariants of the linear approximation of the considering system. As a specific example, we considered the approximation for the system of conservation laws for shallow water theory.
To determine the accuracy of the scheme in the regions influenced by shocks that propagate with variable velocity and behind the fronts of which a non-constant solution is formed, series of test calculations on the sequence of contracting grids were carried out. These calculations enabled us to apply the Runge rule to define the order of convergence of the finite-difference solution. To estimate the accuracy of the modified CABARET scheme, the orders of its integral and local convergence are calculated, as well as local disbalances in calculating of the absolute value of averaged basis functions
vector and local disbalances in calculating of the absolute value of the components of the averaged vector of invariants. The order of the integral convergence makes it possible to estimate the accuracy of finite-difference schemes in translation of the RankineHugoniot conditions across a non-stationary shock wave front. In this case, the order of integral convergence is calculated for the finite-difference solution itself (rather than for its absolute value, as in the L1 norm) on spatial intervals crossing the shock wave.
It is shown that, like TVD-schemes of high order approximation on smooth solutions, the CABARET scheme being formally of the second order, in spite of high accuracy in the localization of shocks, reduces the order of integral convergence to the first order on the integration intervals, one of the boundaries of which is in the region influenced by a shock. The main reason of this decrease in accuracy is that the flux correction used in the CABARET scheme to its monotonization leads to the decrease in the smoothness of the finite-difference fluxes, which in turn leads to the decrease in the order for approximation of Rankine-Hugoniot conditions on the shocks. As a result, the local accuracy of the CABARET scheme in the regions of shocks influence also decreases to about the first order.
Existence and uniqueness of the Riemann problem for a nonlinear system of conservation laws of mixed type
