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**Computation of oscillatory solutions to hyperbolic differential equations using particle methods.**
*(English)*
Zbl 0653.76049

Vortex methods, Proc. UCLA Workshop, Los Angeles/Cal. 1987, Lect. Notes Math. 1360, 68-82 (1988).

[For the entire collection see Zbl 0648.00012.]

Particle method approximations of hyperbolic partial differential equations with oscillatory solutions are studied. Convergence is analyzed in the practical case for which the continuous solution is not accurately resolved on the computational grid. Averaged difference approximations of linear problems and particle method approximations of semilinear problems are presented. Highly oscillatory solutions to the Carleman and Broadwell models as well as the 2-D incompressible Euler equations are considered. The solutions of the continuous and the corresponding numerical models are shown to converge to the same homogenized limit as the frequency in the oscillation increases.

Particle method approximations of hyperbolic partial differential equations with oscillatory solutions are studied. Convergence is analyzed in the practical case for which the continuous solution is not accurately resolved on the computational grid. Averaged difference approximations of linear problems and particle method approximations of semilinear problems are presented. Highly oscillatory solutions to the Carleman and Broadwell models as well as the 2-D incompressible Euler equations are considered. The solutions of the continuous and the corresponding numerical models are shown to converge to the same homogenized limit as the frequency in the oscillation increases.

### MSC:

76N15 | Gas dynamics (general theory) |

76M99 | Basic methods in fluid mechanics |

35L65 | Hyperbolic conservation laws |