Finite difference method

Abstract: A linear model for neutral surface-layer flow over complex terrain is presented. The spectral approach in the two horizontal coordinates and the finite-difference method in the vertical combines the simplicity and computational efficiency of linear methods with flexibility for closure schemes of finite-difference methods. This model makes it possible to make high-resolution computations for an arbitrary distribution of surface roughness and topography. Mixing-length closure as well as E − e closure are applied to two-dimensional flow above sinusoidal variations in surface roughness, the step-in-roughness problem, and to two-dimensional flow over simple sinusoidal topography. The main difference between the two closure schemes is found in the shear-stress results. E − e has a more realistic description of the memory effects in length and velocity scales when the surface conditions change. Comparison between three-dimensional model calculations and field data from Askervein hill shows that in the outer layer, the advection effects in the shear stress itself are also important. In this layer, an extra equation for the shear stress is needed.

Abstract: A stable and explicit second order accurate finite difference method for the elastic wave equation in curvilinear coordinates is presented. The discretization of the spatial operators in the method is shown to be self-adjoint for free-surface, Dirichlet and periodic boundary conditions. The fully discrete version of the method conserves a discrete energy to machine precision.

Abstract: The method of normal modes is frequently used to solve acoustic propagation problems in stratified oceans. The propagation numbers for the modes are the eigenvalues of the boundary value problem to determine the depth dependent normal modes. Errors in the numerical determination of these eigenvalues appear as phase shifts in the range dependence of the acoustic field. Such errors can severely degrade the accuracy of the normal mode representation, particularly at long ranges. In this paper we present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes. It consists of a combination of well‐known numerical procedures such as Sturm sequences, the bisection method, Newton’s and Brent’s methods, Richardson extrapolation, and inverse iteration. We also introduce a modified Richardson extrapolation procedure that substantially increases the speed and accuracy of the computation.