THE FINITE MASS METHOD ∗ CHRISTOPH GAUGER † , PETER LEINEN † , AND HARRY YSERENTANT † SIAM J. NUMER. ANAL. c  2000 Society for Industrial and Applied Mathematics Vol. 37, No. 6, pp. 1768–1799 Abstract. The finite mass method, a new Lagrangian method for the numerical simulation of gas flows, is presented and analyzed. In contrast to the finite volume and the finite element method, the finite mass method is founded on a discretization of mass, not of space. Mass is subdivided into small mass packets of finite extension, each of which is equipped with finitely many internal degrees of freedom. These mass packets move under the influence of internal and external forces and the laws of thermodynamics and can undergo arbitrary linear deformations. The method is based on an approach recently developed by Yserentant and can attain a very high accuracy. Key words. finite mass method, gridless discretizations, compressible fluids AMS subject classifications. 76N99, 76M25, 65M99 PII. S0036142999352564 1. Introduction. Fluid mechanics is usually stated in terms of conservation laws that link the change of a quantity like mass or momentum inside a given volume to a flux of this quantity across the boundary of the volume. The finite volume method is directly based on this formulation. Space is subdivided into little cells, and the balance laws for mass, momentum, and energy are set up for each of these cells separately. Similarly, the finite element method is based on a discretization of space and a choice for the trial functions on the resulting cells. In contrast, the finite mass method is founded on a discretization of mass, an idea which is at least as obvious and that can be traced back to the work of von Neumann [10] or Pasta and Ulam [9] in the late 1940s and the 1950s. Instead of dividing space into elementary cells, we divide mass into a finite number of mass packets of finite extension, each of which is equipped with a given number of internal degrees of freedom. These mass packets move under the influence of internal and external forces and the laws of thermodynamics and can intersect and penetrate each other. They can contract, expand, rotate, and even change their shape. Their internal mass distribution is described by a fixed shape function, similarly to finite elements. Although the finite mass method is a purely Lagrangian approach, it has not much to do with particle methods as they are used for Boltzmann-like transport equations; in some way, it is much closer to finite element and finite volume schemes. The approximations it produces are differentiable functions and not discrete measures. The method is basically of second order, and in some experiments we observed even fourth order convergence! The Lagrangian form of description of fluid flows can have many advantages. For example, there are no problems with free surfaces and no convection terms arise. Such features make numerical methods based on the Lagrangian view especially attractive for free flows in unbounded space. In fact, one of the most popular methods of this type, Monaghan’s smoothed particle hydrodynamics [8], has its origins in astrophysics. Like the smoothed particle hydrodynamics, the finite mass method is a completely * Received by the editors February 16, 1999; accepted for publication (in revised form) November 3, 1999; published electronically May 23, 2000. http://www.siam.org/journals/sinum/37-6/35256.html † Mathematisches Institut der Universit¨at T¨ ubingen, 72076 T¨ ubingen, Germany (yserentant@na.uni-tuebingen.de). 1768