关于二维径向对称域中逆散射问题的研究

Finite element method for radially symmetric solution of a multidimensional semilinear heat equation

This study aims to present the error and numerical blow up analyses of a finite element method for computing the radially symmetric solutions of semilinear heat equations. In particular, this study establishes optimal order error estimates in $L^\infty$ and weighted $L^2$ norms for the symmetric and nonsymmetric formulation, respectively. Some numerical examples are presented to validate the obtained theoretical results.

A quasilinear complexity algorithm for the numerical simulation of scattering from a two-dimensional radially symmetric potential

Standard solvers for the variable coefficient Helmholtz equation in two spatial dimensions have running times which grow quadratically with the wavenumber $k$. Here, we describe a solver which applies only when the scattering potential is radially symmetric but whose running time is $\mathcal{O}\left(k \log(k) \right)$ in typical cases. We also present the results of numerical experiments demonstrating the properties of our solver, the code for which is publicly available.

Radially symmetric solutions of the ultra-relativistic Euler equations

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure $p$, the spatial part $\underline{u} \in \R^3$ of the dimensionless four-velocity and the particle density $n$. Radially symmetric solutions of these equations are studied. Analytical solutions are presented for the linearized system. For the original nonlinear equations we design and analyze a numerical scheme for simulating radially symmetric solutions in three space dimensions. The good performance of the scheme is demonstrated by numerical examples. In particular, it was observed that the method …

有界径向对称毛细曲面的多尺度谱方法

Multi-scale spectral methods for bounded radially symmetric capillary surfaces

针对由有界生成曲线描述的径向对称毛细曲面，提出基于切比雪夫多项式插值、结合区域分解与子域自适应细化的多尺度伪谱方法，可适配多种毛细流体场景的边值问题求解。

A mass-lumping finite element method for radially symmetric solution of a multidimensional semilinear heat equation with blow-up

This study presents a new mass-lumping finite element method for computing the radially symmetric solution of a semilinear heat equation in an $N$ dimensional ball ($N\ge 2$). We provide two schemes, (ML-1) and (ML-2), and derive their error estimates through the discrete maximum principle. In the weighted $L^{2}$ norm, the convergence of (ML-1) was at the optimal order but that of (ML-2) was only at sub-optimal order. Nevertheless, scheme (ML-2) reproduces a blow-up of the solution of the original equation. In fact, in scheme (ML-2), we could accurately approximate the blow-up time. Our theor…

A fast adaptive algorithm for scattering from a two dimensional radially-symmetric potential

In the present paper we describe a simple black box algorithm for efficiently and accurately solving scattering problems related to the scattering of time-harmonic waves from radially-symmetric potentials in two dimensions. The method uses FFTs to convert the problem into a set of decoupled second-kind Fredholm integral equations for the Fourier coefficients of the scattered field. Each of these integral equations are solved using scattering matrices, which exploit certain low-rank properties of the integral operators associated with the integral equations. The performance of the algorithm is …