Particle tracking solutions of vector fractional differential equations: A review
-
Yong Zhang
Abstract
This chapter reviews the Lagrangian solvers developed in the last two decades for fractional differential equations (FDEs). Both the Langevin approach and the fractional Lévy motion define dynamics of random walkers, whose density solves the FDE. For the vector FDEs, a multi-scaling compound Poisson process can track the trajectory of particles moving along arbitrary directions with direction-dependent scaling rates. Random walk particle tracking (RWPT) schemes, including streamline projection and flow subordination, are also needed to track particles whose mechanical dispersion follows streamlines. Particle paths affected by boundaries can also be modeled using RWPT, leading to a fully Lagrangian approximation for the vector spatiotemporal FDEs with streamline-dependent superdiffusion in domains of any size and boundary conditions, as required by real-world applications.
Abstract
This chapter reviews the Lagrangian solvers developed in the last two decades for fractional differential equations (FDEs). Both the Langevin approach and the fractional Lévy motion define dynamics of random walkers, whose density solves the FDE. For the vector FDEs, a multi-scaling compound Poisson process can track the trajectory of particles moving along arbitrary directions with direction-dependent scaling rates. Random walk particle tracking (RWPT) schemes, including streamline projection and flow subordination, are also needed to track particles whose mechanical dispersion follows streamlines. Particle paths affected by boundaries can also be modeled using RWPT, leading to a fully Lagrangian approximation for the vector spatiotemporal FDEs with streamline-dependent superdiffusion in domains of any size and boundary conditions, as required by real-world applications.
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations 1
- Time-fractional derivatives 23
- High-order finite difference methods for fractional partial differential equations 49
- Spectral methods for some kinds of fractional differential equations 101
- Spectral methods for fractional differential equations using generalized Jacobi functions 127
- Spectral and spectral element methods for fractional advection–diffusion–reaction equations 157
- Discontinuous Galerkin and finite element methods 185
- Numerical methods for time-space fractional partial differential equations 209
- Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian 249
- Particle tracking solutions of vector fractional differential equations: A review 275
- Singularities 287
- Fast numerical methods for space-fractional partial differential equations 307
- Fast methods for the computation of the Mittag-Leffler function 329
- Index 347
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations 1
- Time-fractional derivatives 23
- High-order finite difference methods for fractional partial differential equations 49
- Spectral methods for some kinds of fractional differential equations 101
- Spectral methods for fractional differential equations using generalized Jacobi functions 127
- Spectral and spectral element methods for fractional advection–diffusion–reaction equations 157
- Discontinuous Galerkin and finite element methods 185
- Numerical methods for time-space fractional partial differential equations 209
- Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian 249
- Particle tracking solutions of vector fractional differential equations: A review 275
- Singularities 287
- Fast numerical methods for space-fractional partial differential equations 307
- Fast methods for the computation of the Mittag-Leffler function 329
- Index 347
