Finite difference method

Abstract: This paper is concerned with the design of a spatial discretization method for polar and nonpolar parabolic equations in one space variable. A new spatial discretization method suitable for use in a library program is derived. The relationship to other methods is explored. Truncation error analysis and numerical examples are used to illustrate the accuracy of the new algorithm and to compare it with other recent codes.

Abstract: The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

Abstract: This paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods. The main focus is on the property of observability, corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary. The mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion, as is often desired in engineering applications. Observability/controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper. However, when the wave equation is discretized by finite difference methods, the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem. In other words, the classical convergence (consistency + stability) property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications. Observability/controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high-frequency spurious solutions for which the group velocity vanishes. This phenomenon is analyzed and several remedies are suggested, including filtering, Tychonoff regularization, multigrid methods, and mixed finite element methods. We also briefly discuss these issues for the heat, beam, and Schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability/controllability properties at the discrete level. We conclude with a list of open problems and future subjects for research.

Abstract: The spontaneously propagating shear crack on a frictional interface has proven to be a useful idealization of a natural earthquake. The corresponding boundary value problems are nonlinear and usually require computationally intensive numerical methods for their solution. Assessing the convergence and accuracy of the numerical methods is challenging, as we lack appropriate analytical solutions for comparison. As a complement to other methods of assessment, we compare solutions obtained by two independent numerical methods, a finite difference method and a boundary integral (BI) method. The finite difference implementation, called DFM, uses a traction-at-split-node formulation of the fault discontinuity. The BI implementation employs spectral representation of the stress transfer functional. The three-dimensional (3-D) test problem involves spontaneous rupture spreading on a planar interface governed by linear slip-weakening friction that essentially defines a cohesive law. To get a priori understanding of the spatial resolution that would be required in this and similar problems, we review and combine some simple estimates of the cohesive zone sizes which correspond quite well to the sizes observed in simulations. We have assessed agreement between the methods in terms of the RMS differences in rupture time, final slip, and peak slip rate and related these to median and minimum measures of the cohesive zone resolution observed in the numerical solutions. The BI and DFM methods give virtually indistinguishable solutions to the 3-D spontaneous rupture test problem when their grid spacing Δx is small enough so that the solutions adequately resolve the cohesive zone, with at least three points for BI and at least five node points for DFM. Furthermore, grid-dependent differences in the results, for each of the two methods taken separately, decay as a power law in Δx, with the same convergence rate for each method, the calculations apparently converging to a common, grid interval invariant solution. This result provides strong evidence for the accuracy of both methods. In addition, the specific solution presented here, by virtue of being demonstrably grid-independent and consistent between two very different numerical methods, may prove useful for testing new numerical methods for spontaneous rupture problems.