Conservative phenomena in continuum mechanics - fluid dynamics, solid mechanics and electromagnetics - are described by Hamiltonian partial differential equations (PDEs). Although the Hamiltonian structure is commonly exploited in the analysis of such equations, the difficulty of maintaining the structure under discretization has resulted in it being virtually neglected in the design of numerical algorithms for continuum mechanics.
The new discipline of geometric integration seeks to preserve mathematical structure in the design of numerical algorithms. Specifically, whereas the traditional approach to numerical modeling views the numerical method as a process that converges to the continuous solution as some discretization parameter h approaches zero, the geometric approach adopts the viewpoint that the numerical method itself is a discrete dynamical system for a fixed value of h and attempts to make that dynamical system model the continuous one by building in exact conservation of structural (geometric) properties, thereby also accurately reproducing the macroscopic conservation laws implied by the differential structure. For Hamiltonian PDEs, the geometry of interest is that of the Hamiltonian (symplectic) structure.
An important question pertains to the proper discrete representation of the Hamiltonian PDE structure. One possibility is to require that the semi-discretization, in space only, yield a Hamiltonian ODE. A more stringent requirement is that symplecticity be conserved locally in both space and time, i.e. that the method be "multisymplectic." Another approach is to investigate the consequences of the Hamiltonian structure in the continuous system, arriving at a set of conservation laws, and then to attack those conservation laws numerically.
The proposed research will attempt to address the above issues by developing geometric methods for Hamiltonian PDEs in continuum mechanics. If successful, this research will provide a fresh and powerful approach to numerical modeling of nondissipative continuum processes, such as ideal fluid dynamics, elastic structural mechanics and electromagnetic fields.
J. Frank and S. Reich, "On spurious reflections, nonuniform grids and finite difference discretizations of wave equations", submitted.
J. Frank and S. Reich,
"The Hamiltonian Particle-Mesh Method for the Spherical
Shallow Water Equations", Atmospheric Science Letters,
5 (2004) 89--95.
C. Cotter, J. Frank and S. Reich,
"Hamiltonian Particle-Mesh Method for Two-Layer Shallow-Water
Equations Subject to the Rigid-Lid Approximation",
SIAM J. Appl. Dyn. Syst., 3 (2004) 69--83.
J. Frank, "Geometric space-time integration of ferromagnetic
materials", Applied Numerical Mathematics, 48 (2004)
307--322.
J. Frank and S. Reich,
"Conservation properties of
smoothed particle hydrodynamics applied to the shallow water
equations", BIT 43 (2003) 40--54.
J. Frank, G. Gottwald and S. Reich, "A Hamiltonian Particle-Mesh Method for the Rotating Shallow Water Equations". Meshfree Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Vol. 26, pp. 131--142, Springer, 2002.
J. Frank and S. Reich, "A Particle-Mesh Method for the Shallow Water Equations Near Geostrophic Balance", J. Comput. Phys. 180 (2002) 407--426.
J. Frank, W. Huang and B. Leimkuhler, "Geometric Integrators for Classical Spin Systems," J. Comput. Phys., 133 (1997) 160--172.