IMPLICIT FINKTE-DIFFERENCE TECHNIQUES FOR SOLUTION OF TWO-DIMENSIONAL EQUATIONS OF MATHEMATICAL PHYSICS BASED ON DIAMOND-TYPE APPROXIMATION AND THEIR APPLICATION IN MATHEMATICAL SIMULATION OF CONTINUOUS MEDIA MECHANICS AND KINETIC PROCESSES

A.D. Gadzhiev, S.Yu. Kuz´min, S.N. Lebedev, V.N. Pisarev, A.A. Shestakov
VANT. Ser.: Mat. Mod. Fiz. Proc 1997. Вып.1. С. 49.

      When coupling simulation of the Lagrangian hydrodynamics with other physical processes, the difference grid becomes significantly nonorthogonal. The efficiency of numerical techniques solving such problems depends on the accuracy of the nonorthogonal difference algorithms applied and on their ability to ensure adequate accuracy on the grids with strong deformations.
      The limitations of traditional nine-dot difference scheme for solving the diffusion-type equations are rather well known at present. The accuracy rapidly decreases when nonorthogonality of a grid increases. This follows from the fact that from among two operators divergence and gradient - the gradient operator is poorly approximated.
      By now some approaches have been proposed for designing difference schemes ensuring adequate accuracy when solving difference equations on nonorthogonal grids. Our approach based on the diamond-type approximation is applicable to a wide class of equations of mathematical physics. In these techniques both divergence and gradient operators are approximated within a preset grid cell, the face values of velocity and pressure being used. Based on the above approach the ROMB-type techniques are easily designed, are efficient, and ensure adequate accuracy on nonorthogonal grids.
      The review considers applications of the ROME technique to:
      — heat conduction equation;
      — system of equations for energies in the three temperature model;
      — gas dynamics equations;
      — hyperbolic systems of general equations;
      — transport equation in both diffusion and P1-approximations;
      — self-conjugate transport equation.
      Results of numerical experiments were presented.