We implement the unified transform method of Fokas as a numerical method to solve linear evolution partial differential equations on the half-line. The method computes the solution at any $x$ and $t$ without spatial discretization or time stepping. With the help of contour deformations and oscillatory integration techniques, the method’s complexity does not increase for large $x,t$ and the method is more accurate as $x,t$ increase (absolute errors are smaller, relative errors are bounded). Our goal is to make no assumptions on the functional form of the initial or boundary functions beyond some decay and smoothness, while maintaining high accuracy in a large region of the $(x,t)$ plane.

中文翻译：

---

半线上初边值问题的数值统一变换方法

我们实现了 Fokas 的统一变换方法作为一种数值方法来求解半线上的线性演化偏微分方程。该方法在任何 $x$ 和 $t$ 处计算解，无需空间离散化或时间步长。借助轮廓变形和振荡积分技术，该方法的复杂度不会随着$x,t$ 的增加而增加，并且随着$x,t$ 的增加该方法更加准确（绝对误差更小，相对误差是有界的）。我们的目标是不对初始函数或边界函数的函数形式做出超出一些衰减和平滑度的假设，同时在 $(x,t)$ 平面的大区域保持高精度。