Adaptative L1 TE and Predictive

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.

References in zbMATH (referenced in 13 articles , 1 standard article )

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  1. Abbaszadeh, Mostafa; Dehghan, Mehdi: Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method (2019)
  2. Ghaffari, Rezvan; Ghoreishi, Farideh: Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations (2019)
  3. Zou, Guang-an: Numerical solutions to time-fractional stochastic partial differential equations (2019)
  4. Liu, Yanzhi; Roberts, Jason; Yan, Yubin: Detailed error analysis for a fractional Adams method with graded meshes (2018)
  5. Li, Yuanlu; Liu, Fawang; Turner, Ian W.; Li, Tao: Time-fractional diffusion equation for signal smoothing (2018)
  6. Xing, Yanyuan; Yan, Yubin: A higher order numerical method for time fractional partial differential equations with nonsmooth data (2018)
  7. Zhou, Zhaojie; Zhang, Chenyang: Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation (2018)
  8. Abbaszadeh, Mostafa; Dehghan, Mehdi: An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate (2017)
  9. Ford, Neville J.; Yan, Yubin: An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data (2017)
  10. Zhokh, Alexey A.; Trypolskyi, Andrey I.; Strizhak, Peter E.: Asymptotic Green’s functions for time-fractional diffusion equation and their application for anomalous diffusion problem (2017)
  11. Angstmann, C. N.; Donnelly, I. C.; Henry, B. I.; Jacobs, B. A.; Langlands, T. A. M.; Nichols, J. A.: From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations (2016)
  12. Moghaderi, Hamid; Dehghan, Mehdi; Hajarian, Masoud: A fast and efficient two-grid method for solving (d)-dimensional Poisson equations (2016)
  13. Yuste, Santos B.; Quintana-Murillo, J.: Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations (2016)