Solution of fluid flow problems by a direct spline interpolation method

A direct cubic spline procedure for the numerical approximation of first and second spatial derivatives is presented. The method uses a uniform mesh. It does not demand the solution of a system of linear equations. The spline coefficients are determined by analytical formulas, depending on the boundary values and the eigenvalues of a 2 X 2 matrix. The method is applied to the solution of the nonlinear one dimensional Burgers equation, the atmospheric boundary layer equations, and to the diffusion equation, where the medium moves at a constant and uniform velocity. Truncation error analysis shows that the formulation is fourth order accurate for the first derivative, and second order accurate for the second derivative if the first derivatives are of fourth order accuracy of boundary points. Stability analysis for the linear Burgers equation, using discrete perturbation theory, gives substantially improved values for the Courant number and the Reynolds cell number compared with the commonly employed spline formulation.