Introduction to “Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov's Method”

✍ Scribed by Ch. Hirsch

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 145 KB
• Volume: 135
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1997.5757

✦ Synopsis

Introduction to ''Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov's Method'' BRAM VAN LEER'S QUEST FOR PERFECTION

B. Van Leer initiated a fundamental analysis of the main properties needing to be satisfied by the ''ultimate'' numer-The paper reproduced in this special issue is the fifth ical schemes, namely monotonicity and conservation couand last of a series, started in 1973 (Refs. [1] to [5]), with pled to second-or higher order accuracy. Within this conthe general objective of a march ''Towards the Ultimate text, the development of Bram's work over the six-year Conservative Difference Scheme.'' This is a unique examduration of his ''ultimate'' series is remarkable. In a free ple of a persistent pursuit, spanning five or six years, of a and creative spirit he undertook a return to the fundamenclearly stated objective, although the outcome was not tals, selecting the notion of monotonicity as the basis of initially known to the author, which led to the foundation the foundation of numerical schemes. Initiating the analyof modern CFD methodology. In commenting on this fifth sis with the LW and Fromm schemes applied to the linear ''installment'' it is difficult to dissociate it from the four one-dimensional convection equation [1, 2], he introduced previous ones, as they form a consistent set, although the in the two initial papers the fundamental concept of slope first four papers of the series deal essentially with linear limiters. Although a similar concept had been introduced, convection. When B. Van Leer started this work, in the at nearly the same time, by Boris and Book [12], Van early seventies, the development of numerical schemes for Leer's approach is distinctive in separating the update prothe compressible flow equations had already reached an cedure into an interpolation or reconstruction step foladvanced stage with the availability of the second-order lowed by an evolution step. This separation greatly clarifies centered scheme of Lax and Wendroff (LW) [6] and its the correct treatment of systems of equations on the basis two-step variant introduced by MacCormack [7, 8]. The of a scalar analysis. This major contribution established latter was a major step forward, since it simplified considerclearly for the first time that the way around the limitations ably the formulation (avoiding computations of the Jacobiexpressed by Godunov's Theorem [14], linking monotoans and requiring only flux evaluations) and opened the nicity to first-order accuracy for linear schemes, was to way to practical applications of the LW scheme, leading introduce nonlinear contributions in the scheme in the to the first significant computations of two-and threeform of limiters. As these limiters require upstream infordimensional, shock-capturing, inviscid and viscous flows mation, it became clear that the way to the ''ultimate'' on complex geometries. Actually, the first practical applicascheme was to look for upwind-biased schemes of at least tion of the finite volume method [9] was based on MacCorsecond-order accuracy. Since the well-known first-order mack's scheme.

upwind CIR scheme [13] cannot be made conservative It was clearly recognised at that time that the secondin a straightforward way, Van Leer turned to the largely order centered LW scheme generated oscillations around ignored work of Godunov [14]. This is another historical shocks. These were accepted as a ''nuisance'' which had merit of the fourth and fifth papers of the series, namely to be filtered out by the addition of higher order dissipation the recognition and extension of Godunov's fundamental terms, or artificial viscosity terms, a concept already intronew approach to numerical methods for hyperbolic conserduced by Von Neumann [10] and also analysed by Lax vation laws, characterized by the introduction of physical, and Wendroff. Based on empirical and intuitive arguments, simple, solutions of the flow equations to the numerical MacCormack and Paullay [11] introduced a more sophistischeme. In the present context, these exact solutions of cated form for the dissipation terms involving a second the inviscid conservation laws describe the time evolution difference of the pressure as a detector of high gradients, of an initial, one-dimensional discontinuity, known as the multiplying second-order differences of the basic variables. Riemann problem. The fourth paper of the series, although This appeared to be very effective essentially, as is known applied to the one-dimensional linear convection equation, today from B. Van Leer's work, because of the nonlinearity sets the basis of modern upwind-biased schemes of secondintroduced hereby.

(or third-) order accuracy, monotone and conservative, Working as an astronomer, with the task of simulating based on piecewise linear, but limited, variations of the the formation of stars and stellar systems, a task somewhat solutions, coupled to the exact solution of the cell interface discontinuities, following Godunov's approach. It also con-isolated from the main pressure of aerospace applications, 227