"Time Stepping in Fluid Dynamics and Other Applications"
Department of Scientific Computing
Florida State University
A key step for successfully solving time dependent PDEs is developing a time integrator. In addition to resolving any stiffness in the PDE, a good time integrator should be high-order, adaptive, and computationally efficient. Many methods discretize in space and then solve a large, but typically sparse, coupled system of ODEs. In contrast, Rothe's method discretizes in time and then solves a time-independent PDE at each time step.
When Rothe's method is applied to the heat equation, the result is the Yukawa equation which must be solved at each time step. I will discuss a high-order and fast method for solving the Yukawa equation, and use it to solve the Allen-Cahn equation and an elastic membrane problem. Then, I will use problems in interfacial dynamics and plasma physics to illustrate how adaptivity, high-order accuracy, and regularizations can be employed.