Formulation and experiments with high‐order compact schemes for nonuniform grids

Abarbanel, S. and Kumar, A. (1988, “Compact higher‐order schemes for the Euler equations”, Journal of Scientific Computing, Vol. 3, pp. 275‐88.

Anderson, D.A., Tannehill, J.C. and Pletcher, R.H. (1984, Computational Fluid Mechanics and Heat Transfer, Hemisphere Publishing Corporation, New York, NY.

Boisvert, R.F. (1981, “Families of high order accurate discretizations of some elliptic problems”, SIAM Journal on Scientific and Statistical Computing, Vol. 2 No. 3, pp. 268‐84.

Brandt, A. (1982, “Guide to multigrid development”, in Hackbush, W. and Trottenberg, V. (Eds), Multigrid Methods: Proceedings, Koln‐Porz, 1981, Springer‐Verlag, New York, NY, pp. 220‐312.

Dennis, S.C.R. and Hudson, J.D. (1989, “Compact h4 finite difference approximations to operators of Navier‐Stokes type”, Journal of Computational Physics, Vol. 85, pp. 390‐416.

Dukowicz, J.K. and Ramshaw, J.D. (1979, “Tensor viscosity method for convection in numerical fluid dynamics”, Journal of Computational Physics, Vol. 32, pp. 71‐9.

Gupta, M.M.et al. (1982), “Single‐cell high order difference method for steady state advection‐diffusion equation”, Proceedings of the Symposium ‐ International Association for Hydraulic Research, pp. 101‐08.

Gupta, M.M., Manohar, R.P. and Stephenson, J.W. (1984, “A single cell high order scheme for the convection‐diffusion equation with variable coefficients”, International Journal for Numerical Methods in Fluids, Vol. 4, pp. 641‐51.

Hemker, P.W. (1982, “Mixed defect correction iteration for the accurate solution of the convection diffusion equation”, in Hackbush, W. and Trottenberg, U. (Eds), Multigrid Methods: Proceedings, Koln‐Porz, 1981, Lecture Notes in Mathematics, Springer‐Verlag, New York, NY, pp. 485‐501.

Johnson, R.W. and MacKinnon, R.J. (1992, “An auxiliary equation method obtaining superconvergent finite element approximations”, Communications in Applied Numerical Methods, Vol. 8, pp. 99‐107.

Leonard, B.P. (1979, “A stable and accurate convective modeling procedure based on quadratic upstream interpolation”, Computer Methods in Applied Mechanics and Engineering, Vol. 19, pp. 59‐98.

Leonard, B.P. and Mokhtari, S. (1990, “Beyond first‐order upwinding: the ultra‐sharp alternative for non‐oscillatory steady‐state simulation of convection”, International Journal for Numerical Methods in Engineering, Vol. 30, pp. 729‐66.

Lynch, R.E. and Rice, J.R. (1978, “High accuracy finite difference approximation to solutions of elliptic partial differential equations”, Proceedings of the National Academy of Sciences, USA, Vol. 75 No. 6, pp. 2541‐4.

MacKinnon, R.J. and Carey, G.F. (1989, “Superconvergent derivatives: a Taylor series analysis”, International Journal for Numerical Methods in Engineering, Vol. 28, pp. 489‐509.

MacKinnon, R.J. and Carey, G.F. (1990, “Nodal superconvergence and solution enhancement for a class of finite element and finite difference methods”, SIAM Journal on Scientific and Statistical Computing, Vol. 11 No. 2, pp. 343‐53.

MacKinnon, R.J. and Johnson, R.W. (1991, “Differential equation based representation of truncation errors for accurate numerical simulation”, International Journal for Numerical Methods in Fluids, Vol. 13, pp. 739‐57.

Noye, B.J. (1990, “New third‐order finite‐difference method for transient one‐dimensional advection‐diffusion”, Communications in Applied Numerical Methods, Vol. 6 No. 4, pp. 279‐88.

Noye, B.J. and Tan, H.H. (1988, “A third‐order semi‐implicit finite difference method for solving the one‐dimensional convection‐diffusion equation”, International Journal for Numerical Methods in Engineering, Vol. 26 No. 7, pp. 1615‐29.

Noye, B.J. and Tan, H.H. (1989, “Finite difference methods for solving the two‐dimensional advection‐diffusion equation”, International Journal for Numerical Methods in Fluids, Vol. 9 No. 1, pp. 75‐98.

Ribbens, C.J. (1989, “A fast adaptive grid scheme for elliptic partial differential equations”, ACM Transaction on Mathematical Software, Vol. 15 No. 3, pp. 179‐97.

Spotz, W.F. (1991, “Superconvergent finite difference methods with applications to viscous flow”, Master’s thesis, University of Texas at Austin, TX.

Spotz, W.F. and Carey, G.F. (1995, “High‐order compact scheme for the stream‐function vorticity equations”, International Journal for Numerical Methods in Engineering, Vol. 38 No. 20, pp. 3497‐512.

Spotz, W.F. and Carey, G.F. (1996, “A high‐order compact formulation for the 3D Poisson equation”, Numerical Methods for Partial Differential Equations, Vol. 12, pp. 235‐43.

White, A. (1989, Personal communication, Los Alamos.