# New efficient algorithms for a general blood tissue transport-metabolism model and stiff differential equations

Wednesday, October 15, 2008 - 4:00pm - 5:00pm

Lind 409

Dexuan Xie (University of Wisconsin)

Fast algorithms for simulating mathematical models of coupled blood-tissue transport and metabolism are critical for the analysis of data on transport and reaction in tissue. This talk will introduce a general blood tissue transport-metabolism model governed by a large system of one-dimensional hyperbolic partial differential equations, and then present a new parallel algorithm for solving it. The key part of the new algorithm is to approximate the model as a group of independent ordinary differential equation (ODE) systems such that each ODE system has the same size as the model and can be integrated independently. The accuracy of the algorithm is demonstrated for solving a simple blood-tissue transport model with an analytical solution. Numerical experiments were made for a large-scale coupled blood tissue transport-metabolism model on a distributed-memory parallel computer and a shared-memory parallel computer, showing the high parallel efficiency of the algorithm.

In the second part of this talk, a well-known implicit Runge-Kutta algorithm called the Radau IIA method will be discussed, which is a favorite stiff ODE solver for the new parallel algorithm. The most time consuming part of the Radau IIA method is to solve a large scale nonlinear algebraic system of stage values. Currently, the widely-used nonlinear solver was still a simplified Newton method proposed by Liniger & Willoughby in 1970. In practice, it may suffer poor convergence problems, forcing the Radau IIA method to select too small step sizes in order to guarantee the convergence. To provide the Radau IIA method with a robust nonlinear solver, this talk will present a new simplified Newton algorithm and discuss its convergence and performance. Numerical results confirm that the new algorithm can have better convergence properties than the current one and can significantly improve the performance of the Radau IIA method.

In the second part of this talk, a well-known implicit Runge-Kutta algorithm called the Radau IIA method will be discussed, which is a favorite stiff ODE solver for the new parallel algorithm. The most time consuming part of the Radau IIA method is to solve a large scale nonlinear algebraic system of stage values. Currently, the widely-used nonlinear solver was still a simplified Newton method proposed by Liniger & Willoughby in 1970. In practice, it may suffer poor convergence problems, forcing the Radau IIA method to select too small step sizes in order to guarantee the convergence. To provide the Radau IIA method with a robust nonlinear solver, this talk will present a new simplified Newton algorithm and discuss its convergence and performance. Numerical results confirm that the new algorithm can have better convergence properties than the current one and can significantly improve the performance of the Radau IIA method.