John Noye

Finite difference techniques for partial differential equations

Publisher Summary Analytical methods of solving partial differential equations are usually restricted to linear cases with simple geometries and boundary conditions. The increasing availability of more and more powerful digital computers has made more common the use of numerical methods for solving such equations, in addition to non-linear equations with more complicated boundaries and boundary conditions. This chapter describes one particular method, the method of numerical finite differences. This method is based on the representation of the continuously defined function τ (x,y,z,t) and its derivatives in terms of values of an approximation τ defined at particular, discrete points called gridpoints. From the appropriate Taylors series expansions of τ about such gridpoints, forward, backward, and central difference approximations to derivatives of τ can be developed to convert the given partial differential equation and its initial and boundary conditions to a set of linear algebraic equations linking the approximations τ defined at the gridpoints.

Numerical Methods for Solving the Transport Equation

Abstract The depth-averaged transport equation, which governs the spread of thermal and chemical pollutants in coastal seas, is described. Methods of numerical solutions based on Eulerian and Lagrangian approaches are outlined. For testing numerical methods a number of exact solutions for special cases and suitable error measures are given. An example of their use in testing three finite difference methods for solving the one-dimensional transport equation is given.

Finite Difference Methods for Solving the One-Dimensional Transport Equation

Abstract Finite difference methods of solving the transport equation, which governs the movement of passive pollutants in the sea, are illustrated using the one-dimensional constant-coefficient advection-diffusion equation. The accuracy of the various finite difference equations are compared by means of their modified equivalent partial differential equations and their wave response properties, and the theoretical findings illustrated by a numerical test. A new third-order three-point implicit method is developed with smaller wave speed errors than conventional methods presently used.

The Numerical Solution of Ordinary Differential Equations: Initial Value Problems

Publisher Summary This chapter discusses the numerical solution of ordinary differential equations. The chapter describes several autonomous systems of differential equations. A method of finding an approximate solution, but only to a single first-order equation, is the graphical method. A direction field may be drawn by evaluating f(x,y) at various points in the x-y plane and drawing a small arrow of slope f(x ,y) from (x, y). The approximate solution is then found by sketching a curve from the point (a, n) such that the arrows are tangential to it. The Taylor series methods considered in the chapter belong to the class of one-step methods because only the values x n , y n are required to calculate y n+1 . A one-step method of solving the initial value problem which satisfies the condition of the uniqueness theorem is said to be convergent if the numerical solution y n approaches the analytic solution y(x n ) at any fixed x n ∈ [a,b] as the step length h tends to zero and y 0 tends to η. The condition that x n , is fixed is necessary because if n is fixed x n = a+ nh would approach a for all n as h→0.

Physical processes and pollution in the waters of Spencer Gulf

Abstract Despite much work during the last decade by scientists studying the effect of tides and winds on the water level variations and circulation in Spencer Gulf, there are still many gaps in the knowledge of the physical processes which occur in these waters. This article surveys known information and points out ways of supplementing this. Circulation and turbulent diffusion are considered in detail because of their effect on pollutant and sediment transport.

Time-Splitting the One-Dimensional Transport Equation

Abstract The time-splitting process has been applied to the one-dimensional transport equation so that diffusive and advective processes may be separately considered using finite difference methods of solution. Explicit, marching and implicit difference methods have been considered and the effect of reversing the order of physical processes has been studied. An improved method of treating boundary values at the half-time level is described.

An Investigation of Open Boundary Conditions for Tidal Models of Shallow Seas

An open boundary condition for circulation models has been developed by Blumberg and Kantha (1985) and applied to a model of the Mid Atlantic Bight forced by meteorological as well as tidal input. In attempts to apply this condition to a model of a shallow coastal sea (Spencer Gulf, South Australia) forced only by tidal input, inaccurate results are obtained unless very small values of the time scale parameter T 1 are used. Turning to the more commonly used open boundary conditions, the Sommerfeld condition with phase velocity calculated using c = √gh produces reasonably accurate results, though the phase error is large. A model using the Orlanski-Sommerfeld condition, in which phase velocity is numerically calculated and tidal motions are separated into inflow and outflow modes with appropriate boundary conditions applied to each, is more accurate. However results are still less accurate than the values obtained using a purely height specified condition — the latter set of results compares very well with observations and is used as the “benchmark” solution in this study. While use of forcing in only the time derivative approximation of the Orlanski-Sommerfeld condition provides marginal improvement in accuracy, adjusting the time level of application of the forcing produces a discretisation which is as accurate as that obtained with the purely height specified condition, in the present case. Orlanski-Sommerfeld discretisations with second order accurate space derivative approximations are developed but they provide no further improvement in results for the present model.

Some explicit three-level finite-difference simulations of advection

Finding an accurate solution to the one-dimensional constant-coefficient advection equation is basic to the problem of modelling, for example, the time-dependent spread of contaminants in fluids. A number of explicit finite-difference methods involving two levels in time have been used for this purpose in the past. Unfortunately, the more accurate ones have very wide computational stencils, making it difficult to obtain approximations near the upstream boundary and at the outflow boundary. For explicit methods, more compact stencils are possible only if they involve three levels in time. Until recently, the second-order leapfrog method has been the only available three-level technique. Using a weighted discretisation on a three-level computational stencil, explicit finite-difference equations of third and fourth order have now been developed. Both methods are more accurate at simulating advection in the absence of shocks, than the explicit two-level methods of corresponding order.

A new method for numerical representation of the land-water boundary in lake circulation models

Finite difference methods generally use rectangular grids in numerical models of lake and tidal motions. A disadvantage of this approach is that first-order spatial errors occur in the values on the computational land-water boundary when the land-water boundary is forced to coincide with grid lines. An alternative method is described in this paper, in which the land-water boundary is approximated using a sequence of oblique piecewise linear segments which slice through the grid elements. Velocity information along the segmented boundary is computed using a slip boundary condition and is subsequently interpolated to nearby computational points of the rectangular grid. Impressive numerical predictions using the new approach are compared with predictions from the traditional stepped boundary model and an analytic solution for the depth-averaged solution of linearized wind-driven flow in a lake.

Comparison of Finite Difference and Galerkin Methods in Modelling Depth-Dependent Tidal Flow in Channels

This paper compares two classes of numerical techniques—finite differences and the Galerkin approximation for vertical integration in order to investigate their relative performance in solving linear depth-dependent channel flow. Two finite difference models are considered: one a modified Leap-frog method and the other a forward time method with an Improved Euler time correction, both of which use implicit differencing in the vertical. Three Galerkin based methods (Galerkin-cosine, -Chebyshev, -Legendre) using simple versions of forward time marching methods are also used. These techniques are compared for three types of vertical eddy viscosity profile: constant, asymmetric with a maximum at the surface, and symmetric with a maximum at mid-depth. The effects of changing the spacing of the vertical levels in the finite difference models and choosing different basis functions for the Galerkin-spectral models are particularly investigated. The relative accuracy of the models is determined by comparison with analytic or semi-analytic solutions. With a no-slip bottom boundary condition, the finite difference models are generally more accurate than the Galerkin models. Only in the case of constant eddy viscosity, when cosine functions become eigenfunctions, does the Galerkin approach give more accurate results than the finite difference methods. As the bottom eddy viscocity of the symmetric profiles is reduced, very small grid-spacings are required near the bottom to maintain the accuracy in the case of the finite difference models, but the accuracy of all the Galerkin models rapidly deteriorates. The no-slip condition imposed at the bottom limits the accuracy of the Galerkin methods using the global basis functions.