Numerical solutions to wave-type PDEs utilizing method-of-lines require the ODE solver’s stability domain to include a large stretch of the imaginary axis surrounding the origin. We show here that extrapolation based solvers of Gragg–Bulirsch–Stoer type can meet this requirement. Extrapolation methods utilize several independent time stepping sequences, making them highly suited for parallel execution. Traditional extrapolation schemes use all time stepping sequences to maximize the method’s order of accuracy. The present method instead maintains a desired order of accuracy while employing additional time stepping sequences to shape the resulting stability domain. We optimize the extrapolation coefficients to maximize the stability domain’s imaginary axis coverage. This yields a family of explicit schemes that approaches maximal time step size for wave propagation problems. On a computer with several cores we achieve both high order and fast time to solution compared with traditional ODE integrators.

利用线法的波浪型PDE的数值解需要ODE求解器的稳定性域包括围绕原点的大范围的假想轴延伸。我们在这里表明，基于外推的Gragg–Bulirsch–Stoer类型的求解器可以满足此要求。外推方法利用几个独立的时间步长序列，使其非常适合并行执行。传统的外推方案使用所有时间步长序列来最大化方法的准确性顺序。相反，本方法在使用额外的时间步进序列来形成所得到的稳定性域的同时，保持了期望的精度顺序。我们优化外推系数以最大化稳定域的虚轴覆盖范围。这产生了一系列显式方案，其对于波传播问题接近最大时间步长。与传统的ODE集成商相比，在具有多个内核的计算机上，我们可以同时实现高阶和快速的解决方案。