A conservative numerical method for a time fractional diffusion equation

Geometric numerical integration, the branch of numerical analysis with the goal of finding approximate solutions of differential equations that preserve some structure of the continuous problem, is a well established field of research [5]. In particular, requiring that invariants or conservation laws are preserved, on one hand, applies on the approximations some constraints that are satisfied also by the exact solutions. On the other hand, it guarantees a better propagation of the error over long integration times [3]. In the last two decades, new techniques for finding conservation laws of fractional differential equations have been derived by suitably generalising methods for PDEs [4, 6]. However, the numerical preservation of conservation laws of time fractional differential equations is a research topic still at an embryonic state. This talk deals with the numerical solution of diffusion equations in the form D^α_t u = D^2_x K(u), α ∈ R, where D_x is the partial derivative in space, K is an arbitrary regular function, and D^α_t denotes the Riemann-Liouville fractional derivative of order α. The proposed numerical method combines a finite difference scheme in space with a spectral time integrator and preserves discrete versions of the conservation laws of the original differential equation [1, 2]. The conservative and convergence properties of the proposed method are verified by the computational solution of some numerical experiments. References [1] K. Burrage, A. Cardone, R. D’Ambrosio, B. Paternoster. Numerical solution of time fractional diffusion systems. Appl. Numer. Math., 116 (2017), 82–94. [2] A. Cardone, G. Frasca-Caccia. Numerical conservation laws of time fractional diffusion PDEs. arXiv.2203.01966, (2022). [3] A. Dur´an, J. M. Sanz-Serna. The numerical integration of relative equilibrium solutions. Geometric theory. Nonlinearity, 11, 1547–1567, (1998). [4] G. S. F. Frederico, D. F. M. Torres. Fractional conservation laws in optimal control theory. Nonlinear Dyn., 53 (2008), 215–222. [5] E. Hairer, C. Lubich, G. Wanner. Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations, volume 31 of Springer Series in Computational Mathematics. Springer, Berlin, second edition, 2006. [6] S. Y. Lukashchuk. Conservation laws for time-fractional subdiffusion and diffusionwave equations. Nonlinear Dyn., 80 (2015), 791–802