A Eulerian-Lagrangian numerical scheme for the dispersion-convection equation using conjugate space-time grids

Abstract A new numerical scheme is proposed for the dispersion-convection equation which combines the utility of a fixed grid in Eulerian coordinates with the computational power of the Lagrangian method. Convection is formally decoupled from dispersion in a manner which does not leave room for ambiguity. The resulting convection problem is solved by the method of characteristics on a grid fixed in space. Dispersion is handled by finite elements on a separate grid which may, but need not, coincide wit the former at selected points in spacetime. Information is projected from one grid to another by local interpolation. The conjugate grid method is implemented by linear finite elements in conjunction with piecewise linear interpolation functions and applied to five problems ranging from predominant dispersion to pure convection. The results are entirely free of oscillations. Numerical dispersion exists but can be brought under control either by reducing the spatial increment, or by increasing the time step size, of the grid used to solve the convection problem. Contrary to many other methods, best results are often obtained when the Courant number exceeds 1.

[1]  R. B. Lantz.  Quantitative Evaluation of Numerical Diffusion (Truncation Error) , 1971 .

[2]  David L. Book,et al.  Flux-corrected transport II: Generalizations of the method , 1975 .

[3]  Richard S. Varga,et al.  Numerical methods of higher-order accuracy for diffusion-convection equations , 1968 .

[4]  D. W. Peaceman,et al.  Numerical Calculation of Multidimensional Miscible Displacement by the Method of Characteristics , 1964 .

[5]  P Jamet,et al.  Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements , 1977 .

[6]  P. L. T. Brian,et al.  Numerical solution of convective transport problems , 1963 .

[7]  M. R. Todd,et al.  Methods for Increased Accuracy in Numerical Reservoir Simulators , 1972 .

[8]  Narayan M. Chaudhari,et al.  An Improved Numerical Technique for Solving Multi-Dimensional Miscible Displacement Equations , 1971 .

[9]  Bruce A. Finlayson,et al.  Solution of the transport equations using a moving coordinate system , 1980 .

[10]  G. Pinder,et al.  A Numerical Technique for Calculating the Transient Position of the Saltwater Front , 1970 .

[11]  U. Shamir,et al.  Numerical solutions for dispersion in porous mediums , 1967 .

[12]  J. Bredehoeft,et al.  Computer model of two-dimensional solute transport and dispersion in ground water , 1978 .

[13]  M. Ciment,et al.  Review. The Operator Compact Implicit Method for Parabolic Equations , 1978 .

[14]  P Jamet,et al.  A second order finite element method for the one‐dimensional Stefan problem , 1974 .

[15]  A finite element method for the diffusion-convection equation with constant coefficients , 1978 .

[16]  G. Smith,et al.  Numerical Solution of Partial Differential Equations: Finite Difference Methods , 1978 .

[17]  S. P. Neuman,et al.  Mixed explicit‐implicit iterative finite element scheme for diffusion‐type problems: II. Solution strategy and examples , 1977 .

[18]  B. O'CONNOR,et al.  Rayleigh-Ritz and Galerkin finite elements for diffusion-convection problems , 1973 .

[19]  George F. Pinder,et al.  Simulation of two‐dimensional contaminant transport with isoparametric Hermitian finite elements , 1977 .

[20]  Analysis of some dispersion corrected numerical schemes for solution of the transport equation , 1978 .

[21]  Richard S. Varga,et al.  Application of Oscillation Matrices to Diffusion-Convection Equations , 1966 .

[22]  O. C. Zienkiewicz,et al.  An ‘upwind’ finite element scheme for two‐dimensional convective transport equation , 1977 .

[23]  E. Bresler,et al.  Simultaneous transport of solutes and water under transient unsaturated flow conditions , 1973 .

[24]  Mixed explicit-implicit iterative finite element scheme for diffusion-type problems: I. Theory , 1977 .

[25]  W. Gray,et al.  An analysis of the numerical solution of the transport equation , 1976 .

[26]  D. D. Laumbach.  A High-Accuracy Finite-Difference Technique for Treating the Convection-Diffusion Equation , 1975 .

[27]  George F. Pinder,et al.  Mass transport in flowing groundwater , 1973 .

[28]  P Jamet,et al.  Numerical solution of the eulerian equations of compressible flow by a finite element method which follows the free boundary and the interfaces , 1975 .

[29]  W. Finn,et al.  Finite elements incorporating characteristics for one-dimensional diffusion-convection equation , 1980 .

[30]  Todd F. Dupont,et al.  Development and Application of Variational Methods for Simulation of Miscible Displacement in Porous Media , 1977 .