In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem (IBVP) are treated by the simplified inverse Lax-Wendroff (SILW) procedure. For the fully discrete case, a third order explicit Runge-Kutta method is used as an example for the analysis. Stability is analyzed by both the GKS (Gustafsson, Kreiss and Sundstr\"om) theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

We consider the problem of reconstructing unknown inclusions inside a thermal conductor from boundary measurements, which arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the unknown inclusion and gave its rigorous mathematical justification. In this paper, we continue our previous works and provide a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we analyze the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function of an interior transmission problem for the inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map. Our new findings reveal the essence of the reconstruction method. A convergence result for noisy measurement data is also proved. Second, based on the heat layer potential argument, we perform a numerical implementation of the reconstruction method for the homogeneous inclusion case. Numerical results are presented to show the efficiency and stability of the proposed method.

This paper is concerned with an interior penalty discontinuous Galerkin (IPDG) method based on a flexible type of non-polynomial local approximation space for the Helmholtz equation with varying wavenumber. The local approximation space consists of multiple polynomial-modulated phase functions which can be chosen according to the phase information of the solution. We obtain some {approximation} properties for this space and \textit{a prior} $L^2$ error estimates for the \textit{h}-convergence of the IPDG method using duality argument. We also provide ample numerical examples to show that, building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces.

We consider an inverse time-dependent source problem governed by a distributed time-fractional diffusion equation using interior measurement data. Such a problem arises in some ultra-slowdiffusion phenomena in many applied areas. Based on the regularity result of the solution to the direct problem, we establish the solvability of this inverse problem as well as the conditional stability in suitable function space with a weak norm. By a variational identity connecting the unknown time-dependent source and the interior measurement data, the conjugate gradient method is also introduced to construct the inversion algorithm under the framework of regularizing scheme. We show the validity of the proposed scheme by several numerical examples.

With rheology applications in mind, we present a fast solver for the time-dependent effective viscosity of an infinite lattice containing one or more neutrally buoyant smooth rigid particles per unit cell, in a two-dimensional Stokes fluid with given shear rate. At each time, the mobility problem is reformulated as a 2nd-kind boundary integral equation, then discretized to spectral accuracy by the Nyström method and solved iteratively, giving typically 10 digits of accuracy. Its solution controls the evolution of particle locations and angles in a first-order system of ordinary differential equations. The formulation is placed on a rigorous footing by defining a generalized periodic Green’s function for the skew lattice. Numerically, the periodized integral operator is split into a near image sum— applied in linear time via the fast multipole method—plus a correction field solved cheaply via proxy Stokeslets. We use barycentric quadratures to evaluate particle interactions and velocity fields accurately, even at distances much closer than the node spacing. Using first- order time-stepping we simulate, for example, 25 ellipses per unit cell to 3-digit accuracy on a desktop in 1 hour per shear time. Our examples show equilibration at long times, force chains, and two types of blow-ups (jamming) whose power laws match lubrication theory asymptotics.