Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique

Viscous flow solutions with a cubic spline approximation

A numerical solution of the Burgers' equation using cubic B-splines

A meshfree method for the numerical solution of the RLW equation

This paper presents techniques for the numerical solution of partial differential equations using cubic spline collocation. The main spline relations are presented and incorporated into solution procedures for partial differential equations. The computational algorithm in every case is a tridiagonal matrix system amenable to efficient inversion methods. Truncation errors and stability are briefly discussed. Finally, some examples of their application to parabolic and hyperbolic systems with mixed boundary conditions are presented. The results obtained are encouraging and justify further research in this field.

Numerical integration of partial differential equations using cubic splines

A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

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R. A. Graves

Papers

A cubic spline approximation for problems in fluid mechanics