## Asymptotic Analysis and the Numerical Solution of Partial Differential EquationsIntegrates two fields generally held to be incompatible, if not downright antithetical, in 16 lectures from a February 1990 workshop at the Argonne National Laboratory, Illinois. The topics, of interest to industrial and applied mathematicians, analysts, and computer scientists, include singular per |

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### Contents

Singular Perturbations Asymptotic Evaluation of Integrals and Computational Challenges | 3 |

Capture and the Connection Formulas for the Transition across a Separatrix | 17 |

AsymptoticInduced Domain Decomposition | 31 |

An Instrument of AsymptoticNumerical Methods | 33 |

An Asymptotically Induced Domain Decomposition Method for Parabolic Boundary Layer Problems | 55 |

AsymptoticInduced Numerical Methods for Conservation Laws | 75 |

Perturbation Methods and Their Use in Numerical Computations | 97 |

Asymptotic Analysis of Dissipative Waves with Applications to Their Numerical Simulation | 99 |

Evolution to Detonation in a Nonuniformly Heated Reactive Medium | 161 |

Surface Evolution Equations from Detonation Theory | 175 |

An Asymptotic Analysis of the Quantum Liouville Equation | 185 |

Nonlinear Diffusion Equations Computational Results | 197 |

Asymptotic Behavior of Nonlinear Partial Differential Equations | 215 |

Wave Equations | 217 |

Convergence to Steady State of Solutions of Viscous Conservation Laws | 225 |

Toward the Automation of Asymptotic Analysis | 239 |

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algebraic algorithm amplitudes applied approximation asymptotic analysis asymptotic expansions blow-up boundary conditions boundary layer boundary value problem capture coefficients combustion computed conservation laws consider constant convergence coordinate corresponding defined denote derived determined discretization dissipation domain decomposition method dynamics energy example Figure finite difference flow fluid functions Garbey given grid higher-order hybrid solutions initial conditions integral inviscid iteration lattice Boltzmann methods Lattice Gas Methods leading-order limiting solution linear Liouville equation MAPLE matching Math Mathematics National Laboratory nonlinear oscillations numerical methods numerical solution obtained orthogonal parabolic parameter Partial Differential Equations phase physical polynomials potential primitive equation propagation quantum Liouville equation reduced equation region residual correction saddle approach satisfy scales Section separatrix shock layer SIAM singular perturbation slow variation theory solitary pulses solve sonic stability steady solution subdomains surface technique temperature theorem transition transonic traveling wave variable velocity viscous zero