In this paper an iterative method is proposed to solve a partial differential equation (PDE) with free boundary arising from pricing corporate bond with credit grade migration risk. A iterative algorithm is designed to construct two sequences of fixed internal boundary problems, which produce two weak solution sequences. It is proved that both weak solution sequences are convergent. In each iteration step, an implicit-upwind difference scheme is used to solve the fixed internal boundary problem. It is shown that the scheme is stable and first-order convergent. Numerical experiments verify that the limit of the weak solution sequence is the solution of the free boundary problem. This method simplifies the free boundary problem solving, ensures the stability of the discrete scheme and reduces the amount of calculation.

In this paper, a posteriori grid method for solving a time-fractional Black-Scholes equation governing European options is studied. The possible singularity of the exact solution complicates the construction of the discretization scheme for the time-fractional Black-Scholes equation. The $L1$ method on an arbitrary grid is used to discretize the time-fractional derivative and the central difference method on a piecewise uniform grid is used to discretize the spatial derivatives. Stability properties and a posteriori error analysis for the discrete scheme are studied. Then, an adapted a posteriori grid is constructed by using a grid generation algorithm based on a posteriori error analysis. Numerical experiments show that the $L1$ method on an adapted a posteriori grid is more accurate than the method on the uniform grid.

A system of singularly perturbed initial value problems with weak constrained conditions on the coefficients is considered. First the system of second-order singularly perturbed problems is transformed into a system of first-order singularly perturbed problems with integral terms, which facilitates the subsequent stability and a posteriori error analyses. Then a hybrid difference method with the use of interpolating quadrature rules is utilized to approximate the transformed system. Next a posteriori error analysis for the discretization scheme on an arbitrary mesh is presented. A solution-adaptive algorithm based on a posteriori error estimation is devised to generate a posteriori mesh and obtain approximation solution. Finally numerical experiments show a uniform convergence behavior of second-order for the scheme, which improves the previous results and achieves the optimal convergence order under the given discrete scheme.