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.

翻译：本文展示了一种高顺序推迟校正算法,并使用四级限制差分法解决自由和移动边界问题。 通常,当自由边界问题在固定计算网格上得到解决时,解决方案的顺序较低,因为自由边界解决办法的不连续性,即使使用了高顺序方法。我们通过详细错误分析发现,解决方案的趋同顺序可以通过解决连续校正的有限差异系统而提高到第四顺序,因为纠正是从先前计算的低顺序解决办法中得出的。惩罚迭代很快会汇集在一起,初步的猜测很好。我们用几个例子展示了我们的算法的准确性和效率。数字结果显示,我们的算法为解决方案和自由边界地点提供了第四顺序的趋同。我们还在具有挑战性的美国调价问题上测试了我们的算法。我们的算法给出了预期的高顺序趋同。