Parallel algorithms and software for time-dependent systems of nonlinear partial differential equations with an application in computational biology

In this thesis we develop a fully implicit, parallel nonlinear domain decomposition method and software for solving the two dimensional bidomain equations which model the excitation process of the heart. The bidomain model consists of a coupled system of time-dependent nonlinear partial and ordinary differential equations, including both parabolic and hyperbolic types. To solve the system of equations, we use a nonlinearly implicit backward Euler discretization scheme for the time variable, and the resulting large sparse nonlinear algebraic system is solved using a Newton-Krylov-Schwarz algorithm at each time step. Within each Newton iteration, the Jacobian system of linear equations is solved inexactly using the restarted GMRES method with a restricted additive Schwarz preconditioner. In order to reduce the storage and execution time, an incomplete factorization technique is applied to each of the subdomain systems of equations.; The proposed numerical algorithms are implemented on various distributed memory parallel computers using PETSc (Portable, Extensible Toolkit for Scientific Computation) of Argonne National Laboratory. Computational experiments show that our methods and software are robust with respect to physical and mesh parameters, and the nested iterative method is scalable when using large number of processors. In addition, our nonlinearly implicit algorithm allows the use of time steps much larger than other existing methods.