The paper deals with finite-difference schemes for a three-dimensional diffusion equation with $\psi$-Caputo derivative with respect to the time and space variables. Theoretical and experimental estimates of accuracy and performance of implicit and splitting schemes are given. Finite-difference schemes are combined with algorithms that accelerate computations on the base of fixed memory principle and expansion of integral operator kernel into series. Computational experiments are conducted on a test problem that has an analytical solution for the case of the Caputo–Katugampola derivative. In the experiments we focus on the issue of interdependence between accuracy and speed of calculations. Based on the obtained estimates, we present an algorithm for automatic selection of optimal computational scheme.

本文研究了具有以下情况的三维扩散方程的有限差分格式： $\psi$关于时间和空间变量的Caputo导数。给出了隐式和拆分方案的准确性和性能的理论和实验估计。有限差分方案与基于固定存储原理并将积分运算符内核扩展为级数的算法可以加速计算。针对一个测试问题进行了计算实验，该问题具有Caputo–Katugampola衍生物的解析解。在实验中，我们关注于准确性和计算速度之间的相互依赖性问题。基于获得的估计，我们提出了一种自动选择最佳计算方案的算法。