BAIL 2006 Book of Abstracts - Institut für Numerische und ...

J.A. MACKENZIE, A. NICOLA: A Discontinuous Galerkin Moving Mesh Method for
Hamilton-Jacobi Equations
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An alternative adaptive strategy that has worked well for time dependent problems is
to use moving meshes. A useful way to construct an adaptive moving mesh is to regard it
as the image of a uniform mesh covering a computational domain under a time-dependent
transformation that clusters mesh elements towards areas where improved spatial resolution
is required. The transformation is often found through a variational formulation where the
mapping is the minimiser of a functional involving properties of the mesh and the solution
(see e.g. [10]). To improve the stability and smooth the evolution of the moving mesh
Russell, Huang and coworkers [9] [8] found it useful to obtain the mapping as the solution of
parabolic moving mesh PDEs which are modified gradient flow equations for the minimisation
of a suitable mesh functional.
For problems where solution discontinuities exist the correct choice of an adaptivity
criterion, or monitor function, is problematic. It is not unusual to find in the literature that
many grid adaptation criteria are singular at solution discontinuities. To prevent singularities
producing degenerate meshes either some form of smoothing procedure is employed or a
regularised functional is used in the variational formulation.
The aim of this talk is to consider the use of the DG method of Hu and Shu [7] to solve
HJ equation using a moving mesh method based on the solution of MMPDEs. The governing
equation are transformed to include the effect of the movement of the mesh and this is done
in such a way that the conservation properties of the original equation are not lost. The
adaptive mesh is driven by a monitor function which is shown to be non-singular in the
presence of solution discontinuities. To produce an acceptable mesh we smooth the monitor
function before it is used to drive the adaptive procedure. Numerical examples in one and
two dimensions are presented to demonstrate the effectiveness of the adaptive procedure..
References
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[8] W. Huang. Practical aspects of formulation and solution of moving mesh partial differential
equations. Journal of Computational Physics, 171:753–775, 2001.
[9] W. Huang and R. D. Russell. Moving mesh strategy based upon a gradient flow equation
for two-dimensional problems. SIAM Journal on Scientific Computing, 20:998–1015,
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[11] O. Lepsky, C. Hu, and C. W. Shu. Analysis of the discontinuous Galerkin method for
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[12] F. Li and C.W. Shu. Reinterpretation and simplified implementation of a discontinuous
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1209, 2005.
Speaker: MACKENZIE, J.A. 26 BAIL 2006
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