Instabilities in Glimm’s scheme for two systems of mixed type.

*(English)*Zbl 0663.73010The paper deals with approximations to the solution of initial value problems in the two cases: (i) Ericksen’s model for pure shearing motion of an elastic material and (ii) the motion of an elastic string lying in the plane. Glimm’s scheme of approximation based on the solution of the Riemann problem is employed and the region of hyperbolicity in the state space is invariant. Invariance does not however is obtained for smooth solutions and there is thus no convergence of Glimm’s approximation to smooth solutions in general.

Some illustrations are added to understand the behaviour of the approximate solutions. In (i) there is weak convergence when the time step is suitably adjusted so as to permit the collision of slow waves and the parameter CFL equals 0.5. When the time step is fixed, coalescence of a pair of wave speeds in both (i) and (ii) leads to erratic selection of new states in Glimm’s scheme and refining of discretization results in a random distribution of transition times with Poisson statistics at the macroscopic level. There is no convergence either in the strong or weak topology. In (i) the approximations converge in expectation to the smooth solution which traverses the non-hyperbolic regime.

Some illustrations are added to understand the behaviour of the approximate solutions. In (i) there is weak convergence when the time step is suitably adjusted so as to permit the collision of slow waves and the parameter CFL equals 0.5. When the time step is fixed, coalescence of a pair of wave speeds in both (i) and (ii) leads to erratic selection of new states in Glimm’s scheme and refining of discretization results in a random distribution of transition times with Poisson statistics at the macroscopic level. There is no convergence either in the strong or weak topology. In (i) the approximations converge in expectation to the smooth solution which traverses the non-hyperbolic regime.

Reviewer: S.K.Lakshmana Rao

##### MSC:

74B20 | Nonlinear elasticity |

65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |

35M99 | Partial differential equations of mixed type and mixed-type systems of partial differential equations |

74R99 | Fracture and damage |

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65N99 | Numerical methods for partial differential equations, boundary value problems |

35R25 | Ill-posed problems for PDEs |