评估Evans函数：数值方法中的阶数降低
Evaluating the Evans function: Order reduction in numerical methods
AI总结：
针对Evans函数数值计算中的刚性挑战，研究了Magnus展开法中的阶数降低现象，并发现Evans匹配函数中的幸运抵消保持了四阶收敛性。
摘要：
我们考虑Evans函数的数值评估，这是一个类似Wronskian行列式的函数，出现在行波稳定性的研究中。构建Evans函数涉及匹配依赖于谱参数的线性常微分方程的解。随着谱参数的增长，问题变得刚性。因此，以前使用Gauss-Legendre方法解决此类问题；但最近，基于Magnus展开的方法被提出。这里我们广泛研究一般标量Schrödinger算子的刚性区域。我们表明，尽管四阶Magnus方法遭受阶数降低，但在计算Evans匹配函数时的一个幸运抵消意味着最终结果保持了四阶收敛。Gauss-Legendre方法不会遭受阶数降低，但也没有这种抵消，因此它在最终结果中具有相同的收敛阶。最后，我们讨论了两种方法作为谱工具的相对优点。
英文摘要：
We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss--Legendre method has previously been used for such problems; however more recently, methods based on the Magnus expansion have been proposed. Here we extensively examine the stiff regime for a general scalar Schrödinger operator. We show that although the fourth-order Magnus method suffers from order reduction, a fortunate cancellation when computing the Evans matching function means that fourth-order convergence in the end result is preserved. The Gauss--Legendre method does not suffer from order reduction, but it does not experience the cancellation either, and thus it has the same order of convergence in the end result. Finally we discuss the relative merits of both methods as spectral tools.
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