We study the $L^1$-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in $L^1$-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order $L^1$-error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].

We consider the numerical solution of the free boundary Bernoulli problem by employing level set formulations. Using a perturbation technique, we derive a second order method that leads to a fast iteration solver. The iteration procedure is adapted in order to work in the case of topology changes. Various numerical experiments confirm the efficiency of the derived numerical method.

We consider solving linear ill-posed operator equations. Based on a multi-scale decomposition for the solution space, we propose a multi-parameter regularization for solving the equations. We establish weak and strong convergence theorems for the multi-parameter regularization solution. In particular, based on the eigenfunction decomposition, we develop a posteriori choice strategy for multi-parameters which gives a regularization solution with the optimal error bound. Several practical choices of multi-parameters are proposed. We also present numerical experiments to demonstrate the outperformance of the multi-parameter regularization over the single parameter regularization.

For two-dimensional boundary integral equations of the first kind with logarithmic kernels, the use of the conventional boundary element methods gives linear systems with dense matrix. In a recent work [J. Comput. Math., 22 (2004), pp. 287-298], it is demonstrated that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. The numerical experiments also indicate that the proposed numerical methods require less computational time than the conventional ones while the formal rate of convergence can be preserved. The purpose of this work is to establish a stability and convergence theory for this fast numerical method. The stability analysis depends on a decomposition of the coefficient matrix for the collocation equation. The formal orders of convergence observed in the numerical experiments are proved rigorously.

In this paper, we propose an algorithm for isolating real roots of a given univariate spline function, which is based on the use of Descartes' rule of signs and de Casteljau algorithm. Numerical examples illustrate the flexibility and effectiveness of the algorithm.

This paper presents and analyzes a monotone domain decomposition algorithm for solving nonlinear singularly perturbed reaction-diffusion problems of parabolic type. To solve the nonlinear weighted average finite difference scheme for the partial differential equation, we construct a monotone domain decomposition algorithm based on a Schwarz alternating method and a box-domain decomposition. This algorithm needs only to solve linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear discrete problem. The rate of convergence of the monotone domain decomposition algorithm is estimated. Numerical experiments are presented.

In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the $\mathcal O(h^4)$ term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions.

A new algorithm for finding the inverse of a nonsingular scaled factor circulant matrix is presented by the Euclid's algorithm. Extension is made to compute the group inverse and the Moore-Penrose inverse of the singular scaled factor circulant matrix. Numerical examples are presented to demonstrate the implementation of the proposed algorithm.