A fast time domain solver for the equilibrium Dyson equation

We consider the numerical solution of the real time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times, and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph, and the Sachdev-Ye-Kitaev model

A High-Order Solver for the Heat Equation in 1D domains with Moving Boundaries

We describe a fast high-order accurate method for the solution of the heat equation in domains with moving Dirichlet or Neumann boundaries and distributed forces. We assume that the motion of the boundary is prescribed. Our method extends the work of L. Greengard and J. Strain, “A fast algorithm for the evaluation of heat potentials”, Comm. Pure & Applied Math. 1990. Our scheme is based on a time-space Chebyshev pseudo-spectral collocation discretization, which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with the Green’s function for the heat equation. We present numerical results that exhibit up to eighth-order convergence rates. Assuming N time steps and M spatial discretization points, the evaluation of the solution of the heat equation at the same number of points in space-time requires O(NM log M) work. Thus, our scheme can be characterized as “fast”, that is, it is work-optimal up to a logarithmic factor. Key words. Integral equations, spectral methods, Chebyshev polynomials, moving boundaries, heat equation, quadratures, Nyström’s method, collocation methods, potential theory