Oscillation Theory, Computation, and Methods of Compensated Compactness

In this paper we wish to extend the work of McLaughlin-Papanicolaou-Pironneau [11] to compressible flows. Thus we shall first summarize the results for imcompressible fluids then present the current state of numerical simulation of these problems and finally make some preliminary statements on the extension to compressible flows and the possible applications to turbulence and acoustics.

We shall describe several aspects of a general program dealing with oscillations in solutions to nonlinear partial differential equations. The main problem is to describe the relationship between microscopic oscillations and their macroscopic averages, in terms of both the static structure and the dynamic behavior. One general framework is provided by the following setting.

We describe the integrable structure of solutions of the nonlinear Schrodinger (NLS) equation under periodic and quasiperiodic boundary conditions. We focus on those aspects of the exact theory which reveal the behavior of these solutions under perturbations of initial conditions (i.e. linearized instabilities), and the effects of slow modulations in space and time, perhaps in the presence of external perturbations. These results and methods continue the investigations of Ercolani, Flaschka, Forest and McLaughlin [1–7] on Korteweg-deVries (KdV), sine-Gordon (sG) and sinh-Gordon wavetrains. Our purpose here is to document the corresponding features of NLS solutions; the rigorous analysis that underlies this paper derives from [1–7] and will appear in the thesis of Lee [8].

In this paper we describe high-order accurate Godunov-type schemes for the computation of weak solutions of hyperbolic conservation laws that are essentially non-oscillatory. We show that the problem of designing such schemes reduces to a problem in approximation of functions, namely that of reconstructing a piecewise smooth function from its given cell averages to high order accuracy and without introducing large spurious oscillatons. To solve this reconstruction problem we introduce a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy wherever the function is smooth but avoids having a Gibbs-phenomenon at discontinuities.

In a report written in 1944, [2], von Neumann described a computational method for calculating flows admitting shocks with abitrary prescribed initial data. The method was a finite difference discretization of the equations of compressible flow in which space and time derivatives were differenced systematically, and viscosity and heat conduction were set equal to zero. Calculations at the Aberdeen Proving Ground, using punched card equipment, produced approximations that contained post-shock mesh scale oscillations see Fig. 1. V. Neumann suggested that these oscillations represented the heat energy created by the irreversible action of the shock, and that as Δx, Δt tend to zero, the approximate solutions tend in the weak sense to the correct discontinuous solutions of the fluid equations: “These considerations suggest the surmise that (25) is always a valid approximation of the hydrodynamical motion, but with this qualification: “It is not the x_a(t) which approximate the x(q,t) but the averages of the x_a over an interval of sufficient length of contiguous a’-s. The x_a themselves perform oscillations around these averages and these oscillations do not tend to zero, but they make finite contributions to the total energy”.

Here we give a detailed discussion of recent developments, both formal and rigorous, in the theory of weakly nonlinear geometric optics for constructing asymptotic solutions of quasi-linear hyperbolic systems in one and several space variables. This method is the main perturbation technique used in analyzing nonlinear wave motion for hyperbolic systems. The ideas for this method originated in the late 1940’s and early 1950’s in pioneering work of Landau [8], Lighthill [10], and Whitham [18]. However, it is only in very recent work [4], [5], [7], [1] that these methods have been developed through systematic self-consistent perturbation schemes for resonant and nonresonant wave problems in one and several space dimensions. Sections II and III of this paper give a detailed discussion and description of these formal perturbation methods applied to problems in 1-D and multi-D, respectively. The reader can consult the survey in [161 which reviews the literature on weakly nonlinear hyperbolic waves before 1981 and compare this treatment with the one described in sections II and III to see the recent progress in the field in constructing such formal perturbation expansions.

This paper summarizes the status of a direct construction of an asymptotic representation of a modulating multiphase wavetrain for a class of perturbed kdV equations. This class includes the kdV-Burgers equation. The calculations apply on a “boundary” between dispersive and dissipative behavior. The construction proceeds by standard asymptotic methods. The result of the construction is an invariant representation of the reduced equations which permits their diagonalization. While mathematically the construction is incomplete, care is taken to identify the mathematical status of each step in the construction. The equivalence of this constructive approach with postulated averages of conservation laws is established for two phase waves. Finally, the Young measure for this program is constructed explicitly.

A systematic procedure for constructing semi-discrete families of 2m - 1 order accurate, 2m order dissipa-tive, variation diminishing, 2m + 1 point band width, conservation form approximations to scalar conservation laws is presented. Here m is an integer between 2 and 8. Simple first order forward time discretization, used together with any of these approximations to the space derivatives, also results in a fully discrete, variation diminishing algorithm. These schemes all use simple flux limiters, without which each of these fully discrete algorithms is even linearly unstable. Extensions to systems, using a nonlinear field-by-field decomposition are presented, and shown to have many of the same properties as in the scalar case. For linear systems, these nonlinear approximations are variation diminishing, and hence convergent. A new and general criterion for approximations to be variation diminishing is also given. Finally, numerical experiments using some of these algorithms are presented.

The idea of applying the Compensated Compactness theory to hyperbolic systems of conservation laws was originated by L. Tartar [12]. He treated the scalar case (without any information on the derivatives) and proposed a strategy for the 2×2 case.

We shall discuss two applications of the method of compensated compactness. The first application is concerned with the zero dissipative limit and the zero dispersion limit for scalar conservation laws, the former corresponding to the Burgers equation and the latter to the Korteveg de Vries equation

Let (S) be a strictly hyperbolic system of two conservation laws with two unknown functions: L. Tartar [1] studued the parabolic approximation of this system, intending to prove the existence of a weak entropy to the Caucy problem:

This paper is the written version of my lecture delivered at the Institute for Mathematics and its Applications workshop on Oscillation Theory, Computation, and Methods of Compensated Compactness. As both the titles of the workshop and this paper suggest, I believe there is a continuum of ideas and methods relating these topics. Perhaps the unifying word is “regularization” for it is a goal of applied mathematics to understand the analytical and physical meanings of the various regularizations of the conservation laws of continuum mechanics. In particular the ability to pass to the limit as the regularization parameters vanish has been a long standing problem and it seems that major progress has been made on this question recently (indeed by several of the participants of this I.M.A. workshop).

In this talk I wish to outline a procedure for studying the solution of completely integrable evolution equations in a distinguished limit. The procedure is applicable to equations which can be solved by the method of the inverse spectral transformation and have discrete spectral data. The distinguished limit corresponds to the above data tending to a continuum.

We consider the stability of finite-difference approximations to hyperbolic initial-boundary-value problems (IBVPs) in one spatial dimension. A complication is the fact that generally more boundary conditions are required for the discrete problem than are specified for the partial differential equation. Consequently, additional “numerical” boundary conditions are required and improper treatment of these additional conditions can lead to instability and/or inaccuracy. For a linear homogeneous IBVP, a finite-difference approximation with requisite numerical boundary conditions can be written in vector-matrix form as u ^{n+ 1} = Cu ⁿ where C is a matrix operator. Lax-Richtmyer stability requires a uniform bound on C ⁿ (i.e., C to the nth power) in some matrix norm for 0 ≤ t = nΔt ≤ T. One would like to have an algebraic test for Lax-Richtmyer stability. For a matrix C of dimension J (denoted by C _J ), a theorem in linear algebra relates ‖C _J ⁿ ‖ to the spectral radius of C _J as n → ∞, with J fixed. We state a conjecture which extends this theorem to difference approximations for IBVPs where the matrix size J increases linearly with n as n → ∞ which corresponds to mesh refinement in both space and time. The asymptotic behavior of ‖C _J ⁿ ‖ is related directly to the eigenvalues from the von Neumann analysis of the Cauchy problem and the eigenvalues from the normal mode analysis of Gustafsson, Kreiss, and Sundström for the left- and right-quarter plane problems. The conjecture is corroborated by examples where the matrix norm of C _J ⁿ is computed numerically at a fixed time as the mesh is refined. An additional conjecture relates the spectral radius of the matrix C _J as J → ∞ to the spectral radius of an auxiliary Dirichlet problem and the eigenvalues from the normal mode analysis of the left- and right-quarter plane problems.

A one-parameter family of second-order explicit and implicit total variation diminishing (TVD) schemes is reformulated so that a simplier and wider group of limiters is included. The resulting scheme can be viewed as a symmetrical algorithm with a variety of numerical dissipation terms that are designed for weak solutions of hyperbolic problems. This is a generalization of Roe and Davis’s recent works to a wider class of symmetric schemes other than Lax-Wendroff. The main properties of the present class of schemes are that they can be implicit, and, when steady-state calculations are sought, the numerical solution is independent of the time step.