Fast integral equation methods for Rothe’s method applied to the isotropic heat equation

Abstract

a b s t r a c tWe present an efficient integral equation approach to solve the forced heat equation, x, u, t), in a two-dimensional, multiply-connected domain, with Dirichlet boundary conditions. Instead of using an integral equation formulation based on the heat kernel, we discretize in time, first. This approach, known as Rothe's method, leads to a non-homogeneous modified Helmholtz equation that is solved at each time step. We formulate the solution to this equation as a volume potential plus a double layer potential, and both of these potentials are calculated with available tools accelerated by the fast multipole method. For a total of N points in the discretization of the boundary and the domain, the total computational cost per time step is O(N). We demonstrate our approach on the heat equation and the Allen-Cahn equation.

A semi-analytical approach to Green׳s functions for heat equation in regions of irregular shape

“…The term modified is used here because the kernel of the integral representation in (14) is not the fundamental solution of the heat equation (as it is in the classical heat potential) but some Green's function instead. Due to the latter, the form in (14) is not just the exact solution of the heat equation in Ω, but satisfies, in addition, the homogeneous boundary condition of (11).…”

“…The resultant elliptic sub-problems can efficiently be solved then by the classical methods of potential. The works [12][13][14] can serve as a good reference to this approach with Rothe's method and Laguerre transformation used for semidiscretization. Separation of the time variable might also be achieved by solving problem in frequency domain of the Laplace or Fourier transform.…”

Fast integral equation methods for the modified Helmholtz equation

“…This thesis extends the methods outlined in [36] where the authors use Rothe's method to solve (1.1) with an added restriction: they require the forcing term b and the Dirichlet boundary condition be constant at each time step. Extending this work to more general forcing terms and boundary conditions is discussed in Chapter 4.…”

“…For instance, the unsteady Stokes equations [11,13], the unsteady incompressible Navier-Stokes equations [10,38], and the homogeneous heat equation [14] have all been studied. More recently, Rothe's method has been applied to the forced heat equation (1.1) in [36].…”

“…In [36], the authors consider the class of problems where the forcing term g on each Γ k is constant. Let Ω k , k = 1, .…”