James R. Salmon

Bio: James R. Salmon is an academic researcher from Cornell University. The author has contributed to research in topics: Dispersion (optics) & Smoothing spline. The author has an hindex of 3, co-authored 4 publications receiving 124 citations.

Cubic spline boundary elements

Abstract: The boundary integral method is formulated and applied using cubic spline interpolation along the boundary for both the geometry and the primary variables. The cubic spline interpolation has continuous first and second derivatives between elements, thus allowing the accurate calculation of derivative dependent functions (on the boundary) such as velocity in potential flow. The spline functions also smooth the geometry and can represent curved sections with fewer nodes. The results of numerical experiments indicate that the accuracy of the boundary integral equation method is improved for a given number of elements by using cubic spline interpolation. It is, however, necessary to use numerical quadrature. The quadrature slows calculation and/or degrades the accuracy. The numerical experiments indicate that most problems run faster for a given accuracy using linear interpolation. There seems to be a class of problems, however, which requires higher order interpolation and/or continuous derivatives for which the cubic spline interpolation works much better than linear interpolation.

Dispersion analysis in homogeneous lakes

Abstract: This paper is devoted to a comparison of the relative accuracy of two-dimensional vs three-dimensional finite element representations of contaminant dispersion in shallow lakes Formulations of both types are developed, followed by numerical calculations of a hypothetical lake The results indicate that for typical lakes a two-dimensional dispersion analysis can be employed in the absence of a significant advective contribution With significant advection the two-dimensional approach is not sufficiently accurate A two-dimensional dispersion analysis requires approximately the same computational resources as a three-dimensional circulation analysis

Integral Equation Method for linear Water Waves

Abstract: The Boundary Integral Equation Method (BIEM) is applied to transient water wave problems. Only two-dimensional linearized waves are considered. As is general practice, free-surface boundary conditions are applied at the equilibrium surface rather than the actual free surface; thus the problems become fixed-boundary problems rather the free-surface problems. For the cases in which fluid domain is unbounded in the horizontal direction, a radition condition is formulated such that waves pass through the computational boundaries without reflection. The stability limits and frequency distortion of the numerical method are examined and given. Numerical results are compared with analytical solutions or experimental data in three examples. Excellent agreement is observed.

An efficient boundary element method for nonlinear water waves

Abstract: The paper presents a computational model for highly nonlinear 2-D water waves in which a high order Boundary Element Method is coupled with a high order explicit time stepping technique for the temporal evolution of the waves. The choice of the numerical procedures is justified from a review of the literature. Problems of the wave generation and absorption are investigated. The present method operates in the physical space and applications to four different wave problems are presented and discussed (space periodic wave propagation and breaking, solitary wave propagation, run-up and radiation, transient wave generation). Emphasis in the paper is given to describing the numerical methods used in the computation.

Acoustic isogeometric boundary element analysis

Abstract: An isogeometric boundary element method based on T-splines is used to simulate acoustic phenomena. We restrict our developments to low-frequency problems to establish the fundamental properties of the proposed approach. Using T-splines, the computer aided design (CAD) and boundary element analysis are integrated without recourse to geometry clean-up or mesh generation. A regularized Burton–Miller formulation is used resulting in integrals which are at most weakly singular. We employ a collocation-based approach to generate the linear system of equations. The method is verified against closed-form solutions and direct comparisons are made with conventional Lagrangian discretizations. It is demonstrated that the superior accuracy of the isogeometric approach emanates from the exact geometric description encapsulated in the T-spline. The method is then applied to a real-world application to illustrate the potential for integrated engineering design and analysis.

A note on the accuracy of the mild-slope equation

Abstract: The mild-slope equation is a vertically integrated refraction-diffraction equation, used to predict wave propagation in a region with uneven bottom. As its name indicates, it is based on the assumption of a mild bottom slope. The purpose of this paper is to examine the accuracy of this equation as a function of the bottom slope. To this end a number of numerical experiments is carried out comparing solutions of the three-dimensional wave equation with solutions of the mild-slope equation. For waves propagating parallel to the depth contours it turns out that the mild-slope equation produces accurate results even if the bottom slope is of order 1. For waves propagating normal to the depth contours the mild-slope equation is less accurate. The equation can be used for a bottom inclination up to 1:3.

Weighted finite difference techniques for the one-dimensional advection-diffusion equation

Abstract: Various numerical techniques will be developed and compared for solving the one-dimensional advection-diffusion equation with constant coefficient. These techniques are based on the two-level finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The new methods are more accurate and are more efficient than the conventional techniques. These schemes are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared.

High-order compact solution of the one-dimensional heat and advection–diffusion equations

Abstract: In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C 1 -spline collocation method for the resulting linear system of ordinary differential equations. The cubic C 1 -spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O ( h 4 , k 4 ) . Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C 1 -spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.