Adaptative L1 TE and Predictive swMATH ID: 32154 Software Authors: Yuste, Santos B.; Quintana-Murillo, J. Description: The notebook corresponding to this method is Adaptative L1 TE and Predictive.nb: Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations. The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of the number of timesteps. Besides, the solutions of these problems usually involve markedly different time scales, which leads to quite inhomogeneous numerical errors. A natural way to address these difficulties is by resorting to adaptive numerical methods where the size of the timesteps is chosen according to the behaviour of the solution. A key feature of these methods is then the efficiency of the adaptive algorithm employed to dynamically set the size of every timestep. Here we discuss two adaptive methods based on the step-doubling technique. These methods are, in many cases, immensely faster than the corresponding standard method with fixed timesteps and they allow a tolerance level to be set for the numerical errors that turns out to be a good indicator of the actual errors. Homepage: http://www.eweb.unex.es/eweb/fisteor/santos/filesFDE.html Dependencies: Mathematica Keywords: finite difference methods; adaptive methods; fractional diffusion equations; anomalous diffusion; numerical examples; error bounds; algorithm; step-doubling technique Related Software: FODE; ma2dfc; Fractional Heat Conduction Toolbox; Heat Conduction Toolbox; Matplotlib; LFA; Wesseling; Mathematica Cited in: 23 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations. Zbl 1335.65075Yuste, Santos B.; Quintana-Murillo, J. 2016 all top 5 Cited by 45 Authors 4 Yan, Yubin 3 Dehghan Takht Fooladi, Mehdi 3 Li, Hong 3 Liu, Yang 2 Abbaszadeh, Mostafa 2 Liu, Yanzhi 2 Roberts, Jason A. 2 Xu, Qinwu 2 Xu, Yufeng 2 Yin, Baoli 1 Angstmann, Christopher N. 1 Bhardwaj, Akanksha 1 Bohaienko, Vsevolod O. 1 Donnelly, I. C. 1 Elsakout, D. M. 1 Ford, Neville J. 1 Gao, Xinghua 1 Ghaffari, Rezvan 1 Ghoreishi, Farideh 1 Hajarian, Masoud 1 He, Siriguleng 1 Henry, Bruce I. 1 Jacobs, Byron A. 1 Kumar, Alpesh 1 Langlands, T. A. M. 1 Li, Tao 1 Li, Yuanlu 1 Liu, Fawang 1 Moghaderi, Hamid 1 Muttardi, M. M. 1 Nichols, James A. 1 Quintana-Murillo, Joaquín 1 Strizhak, Peter E. 1 Sweilam, Nasser Hassan 1 Tiwari, Awanish Kumar 1 Trypolskyi, Andrey I. 1 Turner, Ian William 1 Xing, Yanyuan 1 Yuste, Santos Bravo 1 Želi, Velibor 1 Zhang, Chenyang 1 Zhokh, Alexey A. 1 Zhou, Zhaojie 1 Zorica, Dusan 1 Zou, Guang’an all top 5 Cited in 13 Serials 6 Numerical Algorithms 3 Journal of Computational Physics 2 Physica A 2 Mathematics and Computers in Simulation 2 Applied Numerical Mathematics 1 Computers & Mathematics with Applications 1 Chaos, Solitons and Fractals 1 Applied Mathematics and Computation 1 International Journal of Computer Mathematics 1 Fractional Calculus & Applied Analysis 1 Communications in Nonlinear Science and Numerical Simulation 1 International Journal of Applied and Computational Mathematics 1 Communications on Applied Mathematics and Computation all top 5 Cited in 12 Fields 19 Numerical analysis (65-XX) 18 Partial differential equations (35-XX) 4 Ordinary differential equations (34-XX) 3 Real functions (26-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Special functions (33-XX) 1 Difference and functional equations (39-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Biology and other natural sciences (92-XX) 1 Information and communication theory, circuits (94-XX) Citations by Year