We consider a general family of nonlocal in space and time diffusion equations with space-time dependent diffusivity and prove convergence of finite difference schemes in the context of viscosity solutions under very mild conditions. The proofs, based on regularity properties and compactness arguments on the numerical solution, allow to inherit a number of interesting results for the limit equation. More precisely, assuming H\"older regularity only on the initial condition, we prove convergence of the scheme, space-time H\"older regularity of the solution depending on the fractional orders of the operators, as well as specific blow up rates of the first time derivative. Finally, using the obtained regularity results, we are able to prove orders of convergence of the scheme in some cases. These results are consistent with previous studies. The schemes' performance is further numerically verified using both constructed exact solutions and realistic examples. Our experiments show that multithreaded implementation yields an efficient method to solve nonlocal equations numerically.

翻译：本文考虑了一类具有时空依赖扩散系数的空间和时间非局部扩散方程，并在极温和条件下证明了有限差分格式在粘性解框架下的收敛性。基于正则性性质和数值解的紧性论证，所得证明可继承极限方程的若干有趣结果。具体而言，仅在初始条件满足Hölder正则性的假设下，我们证明了格式的收敛性、解关于算子分数阶阶数的时空Hölder正则性，以及一阶时间导数的特定爆破速率。最后，利用所得正则性结果，我们能在某些情况下证明格式的收敛阶数。这些结果与已有研究一致。通过构造精确解和实际算例进一步数值验证了格式的性能。实验表明，多线程实现为数值求解非局部方程提供了高效方法。