We present results of the numerical simulation of the transient behavior of shallow junction single photon avalanche diodes (SPAD's). We developed a bidimensional model for above breakdown simulations and show that the initially photogenerated charge density builds up locally by an avalanche multiplication process and then spreads over the entire detector area by a diffusion-assisted process. To model real geometries, we developed a simplified model based on the obtained results. The importance of the photon-assisted spreading mechanism is evaluated and compared with the diffusive one. The contribution of the photon-assisted mechanism is minor in these geometries. The model is compared with the experimental data on the avalanche leading edge and the timing resolution; the agreement is good. We conclude that the model can be considered to be a useful tool for the design of improved structures.

Physics and numerical simulation of single photon avalanche diodes

We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can be employed in an MDA with a resulting computational cost of O(N 2 logN) on an N?×?N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two---point boundary value problems in one space dimension. In many important cases, these eigensystems are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation, finite element Galerkin and spectral methods, and the method of fundamental solutions. For ease of exposition, we focus primarily on the Dirichlet problem for Poisson's equation in the unit square, sketch extensions to other boundary conditions and to more involved elliptic problems, including the biharmonic Dirichlet problem, and report extensions to three dimensional problems in a cube. MDAs have also been used extensively as preconditioners in iterative methods for solving linear systems arising from discretizations of non-separable boundary value problems.

Matrix decomposition algorithms for elliptic boundary value problems: a survey

An implicit difference method for the numerical solution of coupled systems of partial differential equations

Thermal convection phenomena of viscoelastic fluids in a closed-top porous square box heated from below are studied numerically based on a modified Darcy’s law, where two characteristic time constants λ and e are used to describe the viscoelastic properties of the fluid. Results for three set values of λ and e are obtained for a Darcy-Rayleigh number up to 400, and show very different heat transport characteristics from those of Newtonian fluids. Two types of thermal convection are shown to be possible. Oscillatory convection due to the viscoelastic properties of the fluid precedes the onset of steady convection due to thermal boundary instability. After steady convection sets in, it tends to suppress oscillatory convection. The interaction between the two types of thermal convection causes the flow patterns in the porous layer to be more complicated than those of Newtonian fluids. Furthermore, thermal convection of viscoelastic fluids in porous layers undergoes earlier bifurcations with increase of Ra th...

Numerical simulation of thermal convection of a viscoelastic fluid in a porous square box heated from below

In this report we consider block-tridiagonal systems with Toeplitz blocks. Each block is of sizen×n consisting ofn
c×n
c matrices as entries, and there arem×m blocks in the system. The solution of those systems consists of 2n
c
m modified sine transforms and an intermediate solution ofn block-tridiagonal systems. Symmetries in the data vectors are exploited such that one modified sine transform can be computed in terms of one Fourier transform of half the length of the original one, hence requiringO(2.5nlog2
n) operations. Similarly, we only have to solve (n+1)/2 of the intermediate systems due to symmetry.

Morgan Pickering

Papers

An Introduction to Fast Fourier Transform Methods for Partial Differential Equations, with Applications