Topic
Finite difference method
About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.
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01 Jan 2005
TL;DR: The Finite Difference Method (FDM) as discussed by the authorsDM is a 10D tutorial for fitting models to data and dynamic inverse noise, SVD, and LLS for real-time assimilation.
Abstract: The Finite Difference Method.- Introduction.- Finite Difference Calculus.- Elliptic Equations.- Elliptic Iterations.- Parabolic Equations.- Hyperbolic Equations.- The Finite Element Method.- General Principles.- A 10D Tutorial.- Multi-Dimensional Elements.- Time-Dependent Problems.- Vector Problems.- Numerical Analysis.- Inverse Methods.- Inverse Noise, SVD, and LLS.- Fitting Models to Data.- Dynamic Inversion.- Time Conventions for Real-Time Assimilation.- Skill Assessment for Data Assimilative Models.- Statistical Interpolation.- Appendices.- Bibliography.- Index.
106 citations
TL;DR: The STEERB code as mentioned in this paper is based on the solution of three-dimensional modified Fokker-Planck equation, which covers the inner and outer radiation belts with incorporation of the Coulomb collisions, radial diffusion due to magnetic and electric field perturbations, and local pitch angle, energy, and cross-pitch angle-energy diffusion.
Abstract: [1] We present a three-dimensional code (storm-time evolution of electron radiation belt, STEERB) for the storm-time evolution of electron radiation belt following the preceding work. STEERB code is based on the solution of three-dimensional modified Fokker-Planck equation, which covers the inner and outer radiation belts with incorporation of the Coulomb collisions, radial diffusion due to magnetic and electric field perturbations, and local pitch angle, energy, and cross-pitch angle-energy diffusion due to various wave-particle interactions. It is implemented by a split operator technique, in conjunction with the recently developed hybrid finite difference method for local wave-particle interaction, and the fully implicit finite difference method for radial diffusion as well as Coulomb collisions. The resulting numerical model is robust, efficient, and easily parallelizable. Some of the dominant characteristics of electron radiation belt during both quiet and active periods can be well reproduced by STEERB code. Test simulations are performed to evaluate the respective roles of radial diffusion, cyclotron resonant interaction with chorus and plume waves in the global radiation belt dynamics, the sensitivity of numerical results to the uncertainty of wave spectrum, and the importance of cross-pitch angle-energy diffusion in the three-dimensional simulations.
106 citations
TL;DR: The framework needed to apply modern high accuracy numerical methods from computational gas dynamics to this extended system of convection-diffusion equations with stiff source terms is developed.
106 citations
TL;DR: Variations of the Adams--Bashforth, backward differentiation, and Runge--Kutta families of time integrators are considered to solve systems of linear wave equations on uniform, time-staggered grids and are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonStaggered versions of equivalent orders.
Abstract: We consider variations of the Adams--Bashforth, backward differentiation, and Runge--Kutta families of time integrators to solve systems of linear wave equations on uniform, time-staggered grids. These methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders. We investigate the accuracy and stability of these methods analytically, experimentally, and through the use of a novel root portrait technique.
106 citations
TL;DR: New mimetic discretizations of diffusion-type equations (for instance, equations modeling single phase Darcy flow in porous media) on unstructured polygonal meshes are derived and the first order convergence rate and the second-order convergence rate for the pressure are demonstrated with numerical experiments.
Abstract: New mimetic discretizations of diffusion-type equations (for instance, equations modeling single phase Darcy flow in porous media) on unstructured polygonal meshes are derived. The first order convergence rate for the fluid velocity and the second-order convergence rate for the pressure on polygonal, locally refined and non-matching meshes are demonstrated with numerical experiments.
106 citations