Nonconservative hybrid shock capturing schemes.

*(English)*Zbl 0781.65076Hybrid schemes are suggested for a hyperbolic system of conservation laws based on the selection in each node of a difference grid of a suitable approximation of equations via constructed analyzers of the solution smoothness. In the smooth regions, the solution is obtained by easily realized high-order systems based on Runge-Kutta methods with the approximation of convection terms on symmetric stencils. In the regions of discontinuities and other singularities the solution is obtained by more complex non-oscillating schemes with upwind differences approximating the integral laws of conservation in a calculated cell. The resulting hybrid schemes are total variance diminishing schemes (with limited total variance of the numerical solution).

The authors also suggest conservative hybrid schemes based on solution smoothness-dependent choice of the interpolant for the calculation of fluxes on the boundary of the calculated cell. The high accuracy of the resulting algorithms is illustrated by numerical solutions of unidimensional Cauchy problems for the scalar conservation law and system of gas dynamics equations.

The authors also suggest conservative hybrid schemes based on solution smoothness-dependent choice of the interpolant for the calculation of fluxes on the boundary of the calculated cell. The high accuracy of the resulting algorithms is illustrated by numerical solutions of unidimensional Cauchy problems for the scalar conservation law and system of gas dynamics equations.

Reviewer: Y.Shokin (Krasnoyarsk)

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

76N15 | Gas dynamics (general theory) |