Parallel Adaptive Simulation of Multiscale and Multiphysics Phenomena
K. R. Arun
Our current research is about the analysis and numerical simulation of problems governed by a system of conservation laws. These are time-dependent partial differential equations (PDEs), usually hyperbolic and nonlinear, with a particularly simple structure. He studies problems from fluid and gas dynamics, aerodynamics, meteorology and geophysics. The study of numerical solution of hyperbolic conservations laws is an important and interesting field of research because there are special difficulties associated with solving these PDEs, such as shock formation, discontinuous solutions. Numerical methods based on simple finite-difference approximations may behave well for smooth solutions but can give disastrous results when discontinuities and shocks are present. Moreover, physically relevant weak solutions to conservation laws have to be properly characterized and the numerical approximations have to respect this characterization otherwise they would converge to a weak solution which has no physical meaning. Therefore, the study of numerical approximations for nonlinear conservation laws is a challenging task.
Many of the above mentioned problems, particularly those arising in hydraulic, geophysics and meteorology, exhibit a multiscale behaviour due a limiting of characteristic parameters, such as the Mach/Froude number. For such problems, a key step towards efficiency is to build a large timestep algorithm, where the CFL condition is dictated entirely by the flow velocities, not by the fast gravitational or acoustic waves. While first steps towards this goal have been made both for macro/micro explicit timesteps and macro implicit timesteps, a main theoretical challenge is to design macro/macro explicit timesteps. Numerical resolution these multiscale problems and realized via the so called asymptotic preserving schemes which are in general semi-implicit and computationally challenging. In addition, essential for stability are also the following applied mathematics/mathematical modelling issues: multidimensional open boundary conditions; stable treatment of dry states; well-balancing of convective and gravitational forces. Numerical analysis and computer science issues decisive for efficiency are: adaptivity and error control; optimal data structures in 3D. We are currently developing a general purpose, massively parallel adaptive mesh refinement flow solver. The development is demanding a high performance cluster capable of MPI parallelization, supporting some of the freely available adaptive mesh refinement (AMR) libraries, and HDF libraries to handle large amounts of data.