2022/05/02 by Da‐Wei Wang, Wang, Dawei, Kirill Serkh +3
Mathematics · Economics, Econometrics and Finance · #Differential Equations and Numerical Methods #Stochastic processes and financial applications #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2205.00617
This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved on a fixed computational grid, the order of the solution is low due to the discontinuity in the solution at the free boundary, even if a high-order method is used. Using a detailed error analysis, we observe that the order of convergence of the solution can be increased to fourth-order by solving successively corrected finite difference systems, where the corrections are derived from the previously computed lower order solutions. The penalty iterations converge quickly given a good initial guess. We demonstrate the accuracy and efficiency of our algorithm using several examples. Numerical results show that our algorithm gives fourth-order convergence for both the solution and the free boundary location. We also test our algorithm on the challenging American put option pricing problem. Our algorithm gives the expected high-order convergence.