Andrew Majda

High order accurate vortex methods with explicit velocity kernels

Vortex methods of high order accuracy are developed for inviscid, incompressible fluid flow in two or three space dimensions. The velocity kernels are smooth functions given by simple, explicit formulas. Numerical results are given for test problems with exact solutions in two dimensions. It is found that the higher order methods yield a considerably more accurate representation of the velocity field than those of lower order for moderate integration times. On the other hand, the velocity field computed by the point vortex method has very poor accuracy at locations other than the particle trajectories.

Stable viscosity matrices for systems of conservation laws

Abstract A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u 1 + f ( u ) x = 0, uϵR m , is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u t + f ( u ) x = v ( Du x ) x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u 1 + f ′( u 0 ) u x = vDu xx should be well posed in L 2 uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.

The method of fractional steps for conservation laws

SummaryThe stability, accuracy, and convergence of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of scalar conservation laws. In particular, it is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition. Examples of discontinuous solutions are presented where both Strang-type splitting algorithms are only first order accurate but one of the standard first order algorithms is infinite order accurate. Various aspects of the accuracy, convergence, and correct entropy production are also studied when each split step is discretized via monotone schemes, Lax-Wendroff schemes, and the Glimm scheme.

WEAKLY NONLINEAR DETONATION WAVES.

The authors develop a simplified asymptotic model for studying nonlinear detonation waves in chemically reacting fluids which propagate with wave speed close to the acoustical sound speed. In this regime the fluid mechanical and chemical phenomena interact substantially with each other. The model provides simplified equations for describing this interaction. When the model is specialized to unidirectional combustion waves advancing into a region of uniform flow, the travelling waves of this model system moving with positive speed coincide with the travelling waves in a qualitative model for such effects introduced previously by the second author. Some interesting new combustion waves with partial burning are also analyzed. The two main assumptions in deriving the asymptotic model are weak nonlinearity and a sufficiently high activation energy for the chemical kinetics.

Numerical radiation boundary conditions for unsteady transonic flow

A family of numerical boundary conditions for far-field-computational boundaries in calculations involving unsteady transonic flow is devised. These boundary conditions are developed in a systematic fashion from general principles. Both numerical and analytic comparisons with other currently used methods are given.

A systematic approach for correcting nonlinear instabilities

SummaryThe Lax-Wendroff (L-W) difference scheme for a single conservation law has been shown to be nonlinearly unstable near stagnation points. In this paper a simple variant of L-W is devised which has all the usual properties-conservation form, three point scheme, second order accurate on smooth solutions, but which is shown rigorously to beL2 stable for Burgers equation and which is believed to be stable in general. This variant is constructed by adding a simple nonlinear viscosity term to the usual L-W operator. The nature of the viscosity is deduced by first stabilizing, for general conservation laws, a model differential equation derived by analyzing the truncation error for L-W, keeping only terms of order (Δt)2. The same procedure is then carried out for an analogous semi-discrete model. Finally, the full L-W difference scheme is rigorously shown to be stable provided that the C.F.L. restriction λ|ujn|≦0.24 is satisfied.

The validity of the modified equation for nonlinear shock waves

Abstract The modified (model, equivalent) equation is an important tool in designing and analyzing nonlinear difference schemes. In this note, the validity of this principle is rigorously established for nonlinear shock wave solutions and the upwind scheme in a particular case.