We propose a methodology for designing well-balanced numerical schemes to investigate traveling waves in parabolic models from mathematical biology. We combine well-balanced techniques for parabolic models known in the literature with the so-called LeVeque-Yee formula as a dynamic estimate for the spreading speed. This latter formula is used to consider the evolution problem in a moving frame at each time step, where the equations admit stationary solutions, for which well-balanced techniques are suitable. Then, the solution is shifted back to the stationary frame in a well-balanced manner. We illustrate this methodology on parabolic reaction-diffusion equations, such as the Fisher/Kolmogorov-Petrovsky-Piskunov Equation, and a class of equations with a cubic reaction term that exhibit a transition from pulled to pushed waves. We show that the numerical schemes capture in a consistent way simultaneously the wave speed and, to an extent, the so-called Bramson delay.

翻译：我们提出了一种设计适衡数值格式的方法，用于研究数学生物学中抛物线模型的行波。我们将文献中已知的抛物线模型适衡技术与LeVeque-Yee公式相结合，将其作为传播速度的动态估计。该公式被用于在每个时间步考虑移动坐标系中的演化问题，其中方程允许稳态解，使得适衡技术得以适用。随后，解以适衡方式移回固定坐标系。我们在抛物线反应扩散方程上展示了该方法，例如Fisher/Kolmogorov-Petrovsky-Piskunov方程，以及一类具有三次反应项的方程，这些方程表现出从拉拽波到推动波的转变。我们证明，数值格式能够一致地同时捕捉波速，并在一定程度上捕捉所谓的Bramson延迟。