INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, zyxw VOl. 32, 183-205 (1991) AN IMPLICIT FINITE-ELEMENT METHOD FOR HIGH-SPEED FLOWS 0. HASSAN', K. MORGAN' zyxwvu University College, Swansea SA2 BPP, zyxwvu U.K. AND J. PERAIRE Department of Aeronautics, Imperial College, Prince Consort Road, London S W7 2BY, U.K. SUMMARY A fast algorithm is presented for constructing continuous lines, made up of element sides, which pass once through each node of a general unstructured triangular mesh and which are generally aligned in prescribed directions. The lines are used as the basis of an adaptive fully implicit unstructured grid procedure for the solution of two-dimensional problems of steady compressible inviscid and laminar viscous high-speed flows, where the equation system is solved by line relaxation using a block tridiagonal equation solver. For three-dimensional laminar viscous simulations it is proposed to utilize an implicitlexplicit finite-element formulation. In the vicinity of solid walls a grid exhibiting structure in the normal direction is employed while, away from this region, the grid will be totally unstructured. In the structured region, lines in the normal direction to the wall are readily identified, while lines in the surfaces parallel to the solid wall are constructed using the proposed two-dimensional procedure. The implicit algorithm is then used in the structured region and the equation solution is achieved via line relaxation. An explicit form of the solution algorithm is used elsewhere. To illustrate the performance of the proposed method, solutions are obtained for both transonic inviscid and transonic and hypersonic laminar viscous problems in two dimensions. The application of the proposed procedure to the solution of three-dimensional hypersonic laminar viscous flow over a double ellipsoid configuration is also described. INTRODUCTION The use of unstructured grids for the solution of high-speed compressible flows offers the analyst a large degree of flexibility in dealing with complex geometries and readily allows for the incorporation of grid adaptivity. This has already been successfully demonstrated in the context of steady flow simulations employing explicit time stepping schemes. zyx -5 However, the unstructured grid approach has generally been considered to suffer from the disadvantage that it is difficult to construct optimal implicit schemes on general grids and, for this reason, the implicit schemes presented to date have generally utilized point implicit methods br line relaxation strategies on locally structured regions. 7-9 Notable exceptions in this area have been the use of the least squares method for the compressible Euler equations"*" and the Petrov Galerkinlleast-squares approach for the Euler and Navier-Stokes equations, l2 where solution of the implicit equation system has been achieved by the application of conjugate gradient and GMRES algorithms, respectively. zyxwvu ' CDR, Innovation Centre 'Department of Civil Engineering 0029-598 I/ 911090 183-23$11.50 zyxwvu 0 1991 by John Wiley & Sons, Ltd. Received 28 June 1990 Revised 4 October 1990