I. Bohachevsky

Bio: I. Bohachevsky is an academic researcher. The author has contributed to research in topics: Godunov's scheme & Finite difference method. The author has an hindex of 1, co-authored 1 publications receiving 1696 citations.

Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics

Abstract: The method of characteristics used for numerical computation of solutions of fluid dynamical equations is characterized by a large degree of non standardness and therefore is not suitable for automatic computation on electronic computing machines, especially for problems with a large number of shock waves and contact discontinuities. In 1950 v. Neumann and Richtmyer proposed to use, for the solution of fluid dynamics equations, difference equations into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over several mesh points. Then, it was proposed to proceed with the computations across the shock waves in the ordinary manner. In 1954, Lax published the "triangle'' scheme suitable for computation across the shock" waves. A deficiency of this scheme is that it does not allow computation with arbitrarily fine time steps (as compared with the space steps divided by the sound speed) because it then transforms any initial data into linear functions. In addition, this scheme smears out contact discontinuities. The purpose of this paper is to choose a scheme which is in some sense best and which still allows computation across the shock waves. This choice is made for linear equations and then by analogy the scheme is applied to the general equations of fluid dynamics. Following this scheme we carried out a large number of computations on Soviet electronic computers. For a check, some of these computations were compared with the computations carried out by the method of characteristics. The agreement of results was fully satisfactory.

On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws

Abstract: This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is given to the Godunov-type schemes that result from using an approximate solution of the Riemann problem. For schemes based on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionless Boltzmann equation.

Restoration of the contact surface in the HLL-Riemann solver

Abstract: The missing contact surface in the approximate Riemann solver of Harten, Lax, and van Leer is restored. This is achieved following the same principles as in the original solver. We also present new ways of obtaining wave-speed estimates. The resulting solver is as accurate and robust as the exact Riemann solver, but it is simpler and computationally more efficient than the latter, particulaly for non-ideal gases. The improved Riemann solver is implemented in the second-order WAF method and tested for one-dimensional problems with exact solutions and for a two-dimensional problem with experimental results.

Characteristic-based schemes for the euler equations

Abstract: P. L. RoeCollege of Aeronautics, Cranfield Institute of Technology,Cranfield MK43 0AL, EnglandIntroductionComputer simulations of fluid flow provide today the sort of detailedinformation concerning special cases that could previously only beobtained from experime.nts. The computer is attractive as a replacement forexperiments that are difficult, dangerous, or expensive, and as an alternativeto experiments that are impossible. Nevertheless, a computer simulationdoes not have quite the same status as a physical experiment because atpresent there usually remains some doubt about its accuracy. Even thoughthe computer code may be free of error to the extent that it operates exactlyas its author intended, it is seldom possible to give a rigorous proof thatthese intentions were in all respects correct. Most of the practical codeswritten to solve complicated problems contain empirical features, some-times in the form of "adjustable constants" whose values must be "tuned"by appeal to the experiments that the simulations are intended to displace.A computer code is described as being "robust" if it has the virtue ofgiving reliable answers to a wide range of problems without needing to beretuned. The ideal code would be one that fully met some declaredspecification of accuracy and problem range, and whose every line was anecessary contribution to that aim. Few codes yet approach that ideal; amajor impediment is that we presently have little idea what properties canbe specified without contradiction.In recent years, however, our understanding of computer codes for aparticular class of problems has advanced some way toward completeness.The problems are sufficiently complex that naive numerical techniques canproduce disaster, yet sufficiently simple that well-understood physics can3370066-4189/86/0115-0337502.00www.annualreviews.org/aronline Annual Reviews

Computational methods in Lagrangian and Eulerian hydrocodes

Abstract: Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.