I will describe a fast method to compute time convolution integrals, with the goal of numerically solving diffusion-type equations in a way that is computationally efficient in time and memory.

This extends the work of Greengard and Strain on the solution of the heat equation. In particular, I will discuss two aspects of the new algorithm. The first is the choice of quadrature nodes and weights for the Fourier integral representation of the heat kernel and the extension of this discrete representation to fractional powered convolution kernels. The second aspect is the evaluation of heat layer potentials on moving boundaries, which is challenging because time quadrature techniques can be prone to slow convergence due to certain properties of the geometry, even at fine discretizations. I will discuss this problem and our proposed solution.

I will show the application of this approach to the modeling of crystal growth via numerical simulation of the phase field model.