A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two-dimensional unsteady convection-diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter μ and fourth order accurate in space. For 0.5 ≤ μ ≤ 1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection-diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection-diffusion problem and three flows of varying complexities governed by the two-dimensional incompressible Navier-Stokes equations

A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

SUMMARY A new fourth order compact formulation for the steady 2-D incompressible Navier-Stokes equations is presented. The formulation is in the same form of the Navier-Stokes equations such that any numerical method that solve the Navier-Stokes equations can easily be applied to this fourth order compact formulation. In particular in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601×601. Using this formulation, the steady 2-D incompressible flow in a driven cavity is solved up to Reynolds number of 20,000 with fourth order spatial accuracy. Detailed solutions are presented.

Fourth-order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers

In this work we construct an extension to a class of higher-order compact methods for the three-dimensional Poisson equation. A superconvergent nodal rate of O(h6) is predicted, or O(h4) if the forcing function derivatives are not known exactly. Numerical experiments are conducted to verify these theoretical rates. © 1996 John Wiley & Sons, Inc.

A high-order compact formulation for the 3D Poisson equation

In this article, we present an extension of our previous approaches for steady-state higher-order compact (HOC) difference methods to time-dependent problems. The formulation also provides a framework for similar treatment of other HOC spatial schemes. A stability analysis is provided for transient convection-diffusion in 1D and transient diffusion in 2D. Supporting numerical experiments are included to illustrate stability and accuracy as well as oscillatory and dissipative behavior. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 657–672, 2001

Extension of high‐order compact schemes to time‐dependent problems

https://ntrs.nasa.gov/api/citations/19940017115/downloads/19940017115.pdf

Order of accuracy of QUICK and related convection-diffusion schemes

High-order compact finite difference schemes for two-dimensional convection-diffusion-type differential equations with constant and variable convection coefficients are derived. The governing equations are employed to represent leading truncation terms, including cross-derivatives, making the overall O(h4) schemes conform to a 3 × 3 stencil. We show that the two-dimensional constant coefficient scheme collapses to the optimal scheme for the one-dimensional case wherein the finite difference equation yields nodally exact results. The two-dimensional schemes are tested against standard model problems, including a Navier-Stokes application. Results show that the two schemes are generally more accurate, on comparable grids, than O(h2) centred differencing and commonly used O(h) and O(h3) upwinding schemes.

Differential‐equation‐based representation of truncation errors for accurate numerical simulation

A finite-element method is developed which improves accuracy and yields superconvergent approximations to two-dimensional elliptic boundary-value problems on a union of square bilinear elements. This method employs an auxiliary equation which is derived using a Taylor series analysis on the discrete formula obtained from a Galerkin finite-element formulation. Application of the finite-element method to the auxiliary equation results in O(h4) convergence for problems with constant coefficients and improved accuracy for problems with variable coefficients. Supporting numerical examples are given.

An auxiliary equation method for obtaining superconvergent finite‐element approximations

A recent article by Leonard and Mokhtari (1990) discusses the order of accuracy of discrete equations used for numerical simulation. They maintain that discrete operators derived for the finite-volume method should be Taylor-series-analysed using the various points that appear in a finite-volume discrete equation. The current paper argues that any discrete scheme must be analysed from a single point. It is shown that while Leonard's finite-difference QUICK scheme is O(h3)-accurate, his implementation of QUICK for the finite-volume method is only O(h2). An alternative finite-volume formulation is shown to yield a locally third-order-accurate operator in the final discrete equation. Also, third-order-accurate formulae are discussed for various numerical implementations. Numerical examples are included to demonstrate actual rates of convergence for various operators.

Equivalent versions of the quick scheme for finite‐difference and finite‐volume numerical methods

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Differential‐equation‐based representation of truncation errors for accurate numerical simulation

An auxiliary equation method for obtaining superconvergent finite‐element approximations

Equivalent versions of the quick scheme for finite‐difference and finite‐volume numerical methods