M.J. Baines

Moving finite element methods for evolutionary problems. II. Applications

Abstract In this, the second of two papers on the subject, we present applications of the moving finite element method to a number of test problems. Key features are linear elements, a direct approach to parallelism and node overtaking (avoiding penalty functions), rapid inversion of the mass matrix by preconditioned conjugate gradients, and explicit Euler time stepping. The resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy. The paper includes a substantial section on changes of dependent variable and front tracking techniques for non-linear diffusion problems. Test problems include non-linear hyperbolic conservation laws and non-linear parabolic equations in one and two dimensions.

Moving finite element methods for evolutionary problems. I. Theory

Abstract In this paper, the first of two on the subject, we present a unified approach to moving and fixed finite element methods for evolutionary problems in terms of projections. The central theoretical results are concerned with moving finite elements for one-dimensional scalar problems (particularly hyperbolic equations with shocks), but the viewpoint extends to general systems in any number of dimensions.

Grid adaptation via node movement

Abstract The two principal ways of relocating nodes in approximation theory and the numerical solution of partial differential equations (PDEs) are through equidistribution and direct minimization. The benefits and limitations of both approaches are briefly discussed together with implementations which include both functional representation and the solution of steady PDEs. Although equidistribution must be combined with existing numerical schemes when used to adapt the solution of a PDE, direct minimization can be extended automatically to the fairly wide class of PDEs governed by variational principles. Several iteration methods are described for determining optimal grids and solutions in this way, one of which uses the technique of the Moving Finite Element method. Examples are given from different areas of grid adaptation including functional approximation, the solution of Poisson and advection equations, and the shallow water equations. The extension to time-dependent problems is discussed.

A new family of mathematical models describing the human growth curve—Erratum: Direct calculation of peak height velocity, age at take-off and associated quantities

A new family of mathematical functions to fit longitudinal growth data was described in 1978. The ability of researchers to directly use parameters as estimates of age at peak height velocity resulted in them overlooking the possibility of directly calculating these quantities after model estimation. This erratum has corrected three mistakes in the original manuscript in the direct calculation of peak height velocity and age at take-off and has implemented the solutions in a STATA program which directly calculates the estimates, standard errors and confidence intervals for age, height and velocity at peak height velocity.

Multidimensional Least Squares Fluctuation Distribution Schemes with Adaptive Mesh Movement for Steady Hyperbolic Equations

Optimal meshes and solutions for steady conservation laws and systems within a finite volume fluctuation distribution framework are obtained by least squares methods incorporating mesh movement. The problem of spurious modes is alleviated through adaptive mesh movement, the least squares minimization giving an obvious way of determining the movement of the nodes and also providing a link with equidistribution. The iterations are carried out locally node by node, which yields good control of the moving mesh. For scalar equations an iteration which respects the flow of information in the problem significantly accelerates the convergence. The method is demonstrated on a scalar advection problem and a shallow water channel flow problem. For discontinuous solutions we introduce a least squares shock fitting approach which greatly improves the treatment of discontinuities at little extra expense by using degenerate triangles and moving the nodes. Examples are shown for a discontinuous shallow water channel flow and a shocked flow in gasdynamics governed by the compressible Euler equations.

Multidimensional Upwinding with Grid Adaptation

Over the past ten years multidimensional upwinding techniques have been developed with the intention of superseding traditional conservative upwind finite volume methods which rely on the solution of one-dimensional Riemann problems. The new methods attempt a more genuinely multidimensional approach to the solution of the Euler equations by considering a piecewise linear continuous representation of the flow with the data stored at the nodes of the grid. The schemes are then constructed from three separate stages: the decomposition of the system of equations into simple (usually scalar) components, the construction of a consistent, conservative discrete form of the equations and the subsequent solution of the decomposed system using scalar fluctuation distribution schemes. A detailed description of each of these stages can be found in [1, 2, 3].

Data assimilation for morphodynamic prediction and predictability

This paper gives an overview of the project Changing coastlines: data assimilation for morphodynamic prediction and predictability. This project is investigating whether data assimilation could be used to improve coastal morphodynamic modeling. The concept of data assimilation is described, and the benefits that data assimilation could bring to coastal morphodynamic modeling are discussed. Application of data assimilation in a simple ID morphodynamic model is presented. This shows that data assimilation can be used to improve the current state of the model bathymetry, and to tune the model parameter. We now intend to implement these ideas in a 2D morphodynamic model, for two study sites. The logistics of this are considered, including model design and implementation, and data requirement issues. We envisage that this work could provide a means for maintaining up-to date information on coastal bathymetry, without the need for costly survey campaigns. This would be useful for a range of coastal management issues, including coastal flood forecasting.

Moving finite element, least squares, and finite volume approximations of steady and time-dependent PDEs in multidimensions

Abstract We review recent advances in Galerkin and least squares methods for approximating the solutions of first- and second-order PDEs with moving nodes in multidimensions. These methods use unstructured meshes and minimise the norm of the residual of the PDE over both solutions and nodal positions in a unified manner. Both finite element and finite volume schemes are considered, as are transient and steady problems. For first-order scalar time-dependent PDEs in any number of dimensions, residual minimisation always results in the methods moving the nodes with the (often inconvenient) approximate characteristic speeds. For second-order equations, however, the moving finite element (MFE) method moves the nodes usefully towards high-curvature regions. In the steady limit, for PDEs derived from a variational principle, the MFE method generates a locally optimal mesh and solution: this also applies to least squares minimisation. The corresponding moving finite volume (MFV) method, based on the l2 norm, does not have this property however, although there does exist a finite volume method which gives an optimal mesh, both for variational principles and least squares.

A constrained moving finite element solution of the one-dimensional oxygen diffusion with absorption problem

Abstract We derive an adaptive finite-element method for the numerical solution of a one-dimensional, implicit, moving-boundary problem: The oxygen diffusion with absorption problem of Crank and Gupta. The method, which includes a moving grid and incorporates an iteration for the velocity of the moving boundary, produces good results over a range of parameters. We present numerical and graphical results in one dimension and indicate an extension of the method to two dimensions.

A moving-mesh finite difference scheme that preserves scaling symmetry for a class of nonlinear diffusion problems

Abstract A moving-mesh finite difference scheme based on local conservation is presented for a class of scale-invariant second-order nonlinear diffusion problems with moving boundaries that (a) preserves the scaling properties and (b) is exact at the nodes for initial conditions sampled from similarity solutions. Details are presented for one-dimensional problems, the extension to multidimensions is described, and the exactness property is confirmed for two radially symmetric moving boundary problems, the porous medium equation and a simplistic glacier equation. In addition, the accuracy of the scheme is also tested for non self-similar initial conditions by computing relative errors in the approximate solution (in the l ∞ norm) and the approximate boundary position, indicating superlinear convergence.