Moving Mesh Methods For Partial Differential Equation

it moves in time. A popular way to introduce mesh speeds into the mesh functional approach is by use of the so, called gradient flow equations. Following the approach of Huang & Russell a functional is minimised over the computational domain . One way to minimise I is to follow the steepest descent direction given by the first derivative of I. The following 'gradient flow' equations define a flow which converge to the equilibrium state at In practice a modied version of these equations is used in with the inclusion f the familiar correction term and the introduction of P, an operator on the underlying function space. The extra term P is used to choose more suitable directions than that of steepest descent with the terms allowing the user to choose a suitable time scale for the problem. It has already been noted in Section 2.1 that the functional approach in one-dimension can be shown to be equivalent to the equidistribution principle. Moreover the approach here can be shown to be similar to using MMPDE 5 being based on the attracting and repellent forces of the monitor function. Indeed Beckett et al used a similar version of the monitor outlined previously (2.26) in conjunction with a onedimensional analogue of (2.29) for the solution of Burgers' equation. More recently MMPDE 5 has been used in two dimensions as part of an adaptive finite element method by Cao et al for the solution of a combustion problem consisting of coupled nonlinear reaction-diffusion equations. Huang & Russell give multi-dimensional generalisations using this methodology for MMPDE's 4 and 6 Using this approach and the general grid generation functional (2.16), a suitable P is given in terms of the determinants of the two monitor matrices, giving the resulting MMPDE or As with solving for a stationary mesh, the actual computations are carried out after interchanging dependent and independent variables, giving where J is the Jacobian of the coordinate transform. Given this general framework, equivalentMMPDE's can be constructed using the various specific functionals described in Section 2.1 Dirichlet boundary conditions are preferred for the solution of (2.29) as this yields a unique solution, but for many problems this is not applicable since the boundary may not be stationary. Indeed, in some cases it is useful to moves nodes around the fixed boundary, for which many techniques are under investigation, the most popular being preserving a onedimensional arc-length equidistribution of nodes on the boundary (see Huang & Russell, Beckett et al). Huang and Russell, outline a familiar interleaving approach for the solution of the higher dimensional MMPDE combined with the underlying physical PDE as follows :

• Calculate the monitor functions G1 and G2 on the current mesh.

• Update the mesh at time by integrating the MMMPDE(2.29) keeping G1 and G2 constant.

• Integrate the physical PDE to get the

solution at time using the mesh

• Choose a value of for the next time step from the physical PDE.

As with their work in one-dimension. Huang, Ren & Russell suggest that the time correction term is preset by the user or determined by the development of the solution. However the choice of this value in one-dimension is relatively insen, sitive and it is thought to be so in higher dimensions also. Central finite difference discretisations are used by Huang & Russell along with a simple rectangular uniform reference mesh for the computational space. Again, extending the work carried out in one-dimension, the monitor is smoothed locally. On reflection, the functional framework for multidimensional moving mesh methods gathers together all of the work described, both in grid adaption and one-dimensional moving grid techniques, since the strict equidistribution ideas in one-dimension can be written in terms of a functional and the moving mesh methods in higher dimensions are derived from a functional approach to grid adaption. As an interesting aside, work by considers moving mesh methods from a more practical aspect. The authors suggest that many of the moving mesh algorithms before them induce error by not satisfying exactly any relevant conservation laws. Work is continued mesh movement equations are derived for the solution of the Navier Stokes equations from a general scalar quantity conservation law. The fact that relevant physical quantities are conserved almost 'by construction' in the method is considered to be of utmost importance and is the driving force behind the moving grid. Whichever approach is undertaken, a good understanding of the numerical techniques alone may not be good enough for the solution of some problems. We shall continue in the next section by introducing recent work which combines moving mesh methods and self-similar solution techniques, which suggests reasonable choices of monitor functions for certain problems. In particular we shall consider application to the solution of the PME, which we now describe in detail.

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