Nonequilibrium Working Group: Numerical Challenges

The study of nonequilibrium systems poses tremendous numerical challenges. Difficult computational issues such as multiple length and time scales with very large dynamic range, complex geometries, shocks/front-tracking, nonlocal kernels, and the inclusion of fluctuations, can all be encountered while attempting to solve a single nonequilibrium problem. Practical limitations of speed and memory with even the biggest available or planned computers make it imperative to develop new computational strategies in order to solve problems of the scale of difficulty to be tackled in this project. A unique feature of our research program is the very close coupling of analytic methods with the numerical techniques. One of our main aims is to produce better numerical methods by using our knowledge of the underlying physics. As part of this program, new algorithms will be developed for simulating the microscopic and mesoscopic models, modern computational technology such as block-structured adaptive mesh refinement (AMR) for high local resolution, overlapping grids for resolution of complex geometries, and iterative methods appropriate for parallel computers will be incorporated into the numerical models. High performance computing platforms will be used for performing large, detailed simulations of the models as part of the numerical verification phase.

The program of research envisaged by the Working Group requires a Grand Challenge scale commitment of high performance computing resources. The Lab is uniquely suited to this effort because of its large scale computational capabilities and the breadth of its scientific expertise across multidisciplinary lines. The possiblity of implementing this program is a result of the advent of high-end parallel computing platforms, where LANL through its Advanced Computing Laboratory (ACL) has been a recognized leader, harnessing the power of machines such as the CM-200, CM-5, T-3D, and now the SGI/Cray Origin 2000 to the most challenging computational problems in fundamental and applied science.

The simulation of the microscopic models requires accurate and efficient methods for solving the integro-differential equations that describe the evolution of inhomogeneous mean fields self-consistently interacting with internal and external fluctuations and driving forces. Numerical methods for these problems will not be able to rely on the simplifications that result from studying spatially homogeneous mean fields as has been done in the past. We will develop new strategies for dealing with the nonlocal integral kernels. The approximation of alternative equivalent forms for the microscopic equations will also be studied. These include Hamiltonian first-order formulations with integral kernels which are spatially nonlocal, and formulations using non-canonical variables that result in spatially and temporally local, yet overdetermined equations. As with the overall research program, the development of numerical algorithms for these alternative formulations must use techniques that preserve appropriate invariance properties of the original formulations. Thorough numerical verification of these previously untested formulations will also be required. A close synergism between algorithmic development and analysis is essential.

In general, the mesoscale models will be described by nonlinear stochastic PDEs that replace microscopic physics with terms represented by colored, spatially-correlated and/or multiplicative noise. While the models at this level are computationally less demanding than the microscopic models, the numerical methods for this class of problems are less well developed and understood. In tandem with the analytic coarse-graining process for deriving these mesoscale models, the development of numerical methods at this stage must mirror the analytic process, and appropriately preserve the essential aspects of the analytic structure and physics of the microscopic problems. Further development of the numerical methodology for complex nonlinear stochastic PDEs will be a key focus of this project. Among other techniques, generalized symplectic integrators will be developed to preserve constraints under stochastic evolution.

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