Abstract
Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.
Highlights
In this paper, we present a highly accurate numerical method for solving the forced isotropic heat equation with Dirichlet data on complex multiply connected domains in two dimensions
The heat equation is discretised in time with an implicit treatment of the diffusion term, an approach that is sometimes referred to as elliptic marching or Rothe’s method [2, 3]
To solve the heat equation a sequence of modified Helmholtz equations is solved per time step and the number of such problems is set by that specific time stepping scheme
Summary
We present a highly accurate numerical method for solving the forced isotropic heat equation with Dirichlet data on complex multiply connected domains in two dimensions. At that time only examples for which a continuous extension of f could be constructed by hand were considered, excluding complex geometries and general data Another impediment was the loss of accuracy for evaluating layer potentials close to their sources. They solve the Poisson equation with the abovementioned split into a particular and an homogeneous problem We use this method for function extension in the context of the modified Helmholtz equation with excellent results and can consider a larger class of forcing terms compared with [1]. In recent work towards solving the heat equation with said method , Wang et al have developed a hybrid method that allows for evaluation of the boundary and volume potentials including the space-time heat kernel without the constraints from geometric stiffness [22]. The algorithmic development in these efforts is essential to increase
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