Moving Point, Particle, and Free-Lagrange Methods for Convection-Diffusion Equations

Recently proposed moving point and particle numerical methods for time-dependent convection-diffusion equations are briefly reviewed. The methods are applied to a one-dimensional model problem and an alternative and superior method is presented. This is generalised to higher spatial dimensions by reformulating it as a finite-volume or finite-element method on a moving triangulation. It is shown that when solving the heat equation on triangular meshes with Dirichlet data, it is important to use a certain type of triangulation for the methods to possess maximum principles. This triangulation (known as the locally Delaunay triangulation) is shown to be unique. Moreover, it possesses a “minimum-energy” property: hence it may be generated from an arbitrary given triangulation by a sequential process of mesh reconnection. This is important for the convection-diffusion algorithm, as using the transported triangulation from the last time level as an initial guess results in a very efficient mesh generation procedure. The two-dimensional algorithm is thus a free-Lagrange method and is applied to a model problem in two dimensions. Some ideas concerning the generalisation to three dimensions are briefly discussed.

[1]  A. Leonard.  Vortex methods for flow simulation , 1980 .

[2]  Jay P. Boris,et al.  The Lagrangian solution of transient problems in hydrodynamics using a triangular mesh , 1979 .

[3]  W. P. Crowley,et al.  Free-lagrange methods for compressible hydrodynamics in two space dimensions , 1985 .

[4]  Robin Sibson,et al.  Locally Equiangular Triangulations , 1978, Comput. J..

[5]  D. H. McLain,et al.  Two Dimensional Interpolation from Random Data , 1976, Comput. J..

[6]  Robin Sibson,et al.  Computing Dirichlet Tessellations in the Plane , 1978, Comput. J..

[7]  J. M. Nelson.  A triangulation algorithm for arbitrary planar domains , 1978 .

[8]  P. Raviart.  Particle approximation of linear hyperbolic equations of the first order , 1984 .

[9]  J. Monaghan,et al.  Shock simulation by the particle method SPH , 1983 .

[10]  K. Morton.  A comparison of flux limited difference methods and characteristic Galerkin methods for shock modelling , 1987 .

[11]  Christopher R. Anderson,et al.  On Vortex Methods , 1985 .

[12]  C. L. Farmer,et al.  A Moving Point Method for Arbitrary Peclet Number Multi-Dimensional Convection—Diffusion Equations , 1985 .

[13]  D. F. Watson.  Computing the n-Dimensional Delaunay Tesselation with Application to Voronoi Polytopes , 1981, Comput. J..

[14]  Adrian Bowyer,et al.  Computing Dirichlet Tessellations , 1981, Comput. J..

[15]  R W Hockney,et al.  Computer Simulation Using Particles , 1966 .

[16]  J. Donea.  A Taylor–Galerkin method for convective transport problems , 1983 .

[17]  Harold E. Trease,et al.  Three-dimensional free Lagrangian hydrodynamics , 1985 .

[18]  Harold Trease,et al.  The Free-Lagrange Method , 1985 .

[19]  J. Monaghan.  Particle methods for hydrodynamics , 1985 .

[20]  W. Crowley.  A Free-Lagrange method for 2-D compressible flows☆ , 1987 .

[21]  Tohru Ogawa,et al.  A new algorithm for three-dimensional voronoi tessellation , 1983 .

[22]  P. Oelhafen,et al.  About the Delaunay-Voronoi tesselation , 1988 .

[23]  A. H. Armstrong.  A conservative Free Lagrange code , 1985 .

[24]  P. Raviart.  An analysis of particle methods , 1985 .

[25]  B. A. Lewis,et al.  Triangulation of Planar Regions with Applications , 1978, Comput. J..

[26]  P. A. Smith,et al.  Wave-induced bed flows by a Lagrangian vortex scheme , 1985 .

[27]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .