Journal of Computational and Applied Mathematics, ISSN 0377-0427, 2004, Volume 172, Issue 1, pp. 65 - 77

Fractional advection–dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium....

Fractional advection–dispersion | Finite difference approximation | Stability | Radial dispersion | Fractional diffusion | Implicit Euler method | Backward Euler method | Radial advection | Fractional derivative | Numerical fractional ADE | Fractional advection-dispersion | FOKKER-PLANCK EQUATION | MATHEMATICS, APPLIED | WALKS | finite difference approximation | fractional derivative | numerical fractional ADE | fractional advection-dispersion | ANOMALOUS DIFFUSION | ORDER | backward Euler method | TRANSPORT | radial advection | fractional diffusion | implicit Euler method | radial dispersion | LEVY MOTION | stability

Fractional advection–dispersion | Finite difference approximation | Stability | Radial dispersion | Fractional diffusion | Implicit Euler method | Backward Euler method | Radial advection | Fractional derivative | Numerical fractional ADE | Fractional advection-dispersion | FOKKER-PLANCK EQUATION | MATHEMATICS, APPLIED | WALKS | finite difference approximation | fractional derivative | numerical fractional ADE | fractional advection-dispersion | ANOMALOUS DIFFUSION | ORDER | backward Euler method | TRANSPORT | radial advection | fractional diffusion | implicit Euler method | radial dispersion | LEVY MOTION | stability

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 07/2019, Volume 388, pp. 23 - 40

•Derives energy-conserving Galerkin approximation to quasigeostrophy.•Valid for any appropriate and convenient basis.•A particular polynomial basis tested in...

Spectral | Quasigeostrophic | Legendre | Galerkin | EQUATIONS | MODEL | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | SURFACE DYNAMICS | SCALES | EQUILIBRATION | TURBULENCE | MIDOCEAN EDDIES | Energy conservation | Dynamic meteorology | Analysis | Degrees of freedom | Nonlinear equations | Accuracy | Baroclinic instability | Mathematical analysis | Vorticity | Chebyshev approximation | Collocation methods | Galerkin method | Polynomials | Methods | Finite difference method

Spectral | Quasigeostrophic | Legendre | Galerkin | EQUATIONS | MODEL | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | SURFACE DYNAMICS | SCALES | EQUILIBRATION | TURBULENCE | MIDOCEAN EDDIES | Energy conservation | Dynamic meteorology | Analysis | Degrees of freedom | Nonlinear equations | Accuracy | Baroclinic instability | Mathematical analysis | Vorticity | Chebyshev approximation | Collocation methods | Galerkin method | Polynomials | Methods | Finite difference method

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 03/2017, Volume 333, pp. 104 - 127

The Molecular Beam Epitaxial model is derived from the variation of a free energy, that consists of either a fourth order Ginzburg–Landau double well potential...

Energy stability | Second order | Invariant energy quadratization | Molecular beam epitaxial | Unconditional | Linear | THIN-FILM EPITAXY | STABLE SCHEMES | CAHN-HILLIARD EQUATION | SLOPE SELECTION | PHYSICS, MATHEMATICAL | LINEAR SCHEMES | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | ALLEN-CAHN | FINITE-ELEMENT-METHOD | PHASE-FIELD MODEL | DIFFUSE INTERFACE MODEL | 2-PHASE INCOMPRESSIBLE FLOWS | Models | Epitaxy | Analysis | Methods | APPROXIMATIONS | STABILITY | GINZBURG-LANDAU THEORY | EQUATIONS | CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY | THREE-DIMENSIONAL CALCULATIONS | TWO-DIMENSIONAL CALCULATIONS | ACCURACY | COMPUTERIZED SIMULATION | FINITE DIFFERENCE METHOD | MOLECULAR BEAM EPITAXY | SYMMETRY | NONLINEAR PROBLEMS | MOLECULAR BEAMS | FREE ENERGY

Energy stability | Second order | Invariant energy quadratization | Molecular beam epitaxial | Unconditional | Linear | THIN-FILM EPITAXY | STABLE SCHEMES | CAHN-HILLIARD EQUATION | SLOPE SELECTION | PHYSICS, MATHEMATICAL | LINEAR SCHEMES | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | ALLEN-CAHN | FINITE-ELEMENT-METHOD | PHASE-FIELD MODEL | DIFFUSE INTERFACE MODEL | 2-PHASE INCOMPRESSIBLE FLOWS | Models | Epitaxy | Analysis | Methods | APPROXIMATIONS | STABILITY | GINZBURG-LANDAU THEORY | EQUATIONS | CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY | THREE-DIMENSIONAL CALCULATIONS | TWO-DIMENSIONAL CALCULATIONS | ACCURACY | COMPUTERIZED SIMULATION | FINITE DIFFERENCE METHOD | MOLECULAR BEAM EPITAXY | SYMMETRY | NONLINEAR PROBLEMS | MOLECULAR BEAMS | FREE ENERGY

Journal Article

Geophysics, ISSN 0016-8033, 07/2016, Volume 81, Issue 4, pp. C127 - C137

Building anisotropy models is necessary for seismic modeling and imaging. However, anisotropy estimation is challenging due to the trade-off between...

VELOCITY ANALYSIS | GEOCHEMISTRY & GEOPHYSICS | MODELS | MOVEOUT | FINITE-DIFFERENCE CALCULATION | Approximation | Anisotropy | Mathematical analysis | Media | Mathematical models | Aseismic buildings | Estimates | Formulas (mathematics)

VELOCITY ANALYSIS | GEOCHEMISTRY & GEOPHYSICS | MODELS | MOVEOUT | FINITE-DIFFERENCE CALCULATION | Approximation | Anisotropy | Mathematical analysis | Media | Mathematical models | Aseismic buildings | Estimates | Formulas (mathematics)

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 03/2015, Volume 285, pp. 331 - 349

In this paper, the numerical approximation of unsteady convection–diffusion–reaction equations with finite difference method on a special grid is studied in...

Finite element method | Unsteady convection–diffusion–reaction | Finite difference method | Unsteady convection-diffusion-reaction | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RESIDUAL-FREE BUBBLES | STABILIZATION | ELEMENT METHODS | PHYSICS, MATHEMATICAL

Finite element method | Unsteady convection–diffusion–reaction | Finite difference method | Unsteady convection-diffusion-reaction | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RESIDUAL-FREE BUBBLES | STABILIZATION | ELEMENT METHODS | PHYSICS, MATHEMATICAL

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2013, Volume 51, Issue 6, pp. 3458 - 3482

We consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models. Two...

Numerical approximation | Local limit | Finite difference | Discrete maximum principle | Nonlocal diffusion | Peridynamic models | Convergence analysis | Finite element | peridynamic models | MATHEMATICS, APPLIED | VECTOR CALCULUS | discrete maximum principle | finite element | nonlocal diffusion | BOUNDARY-VALUE-PROBLEMS | ELASTICITY | ANOMALOUS DIFFUSION | convergence analysis | MODELS | SOLID MECHANICS | DYNAMICS | NAVIER EQUATION | CONVERGENCE | finite difference | numerical approximation | local limit | FINITE-ELEMENT-METHOD | Numerical analysis | Approximation | Computer simulation | Mathematical analysis | Mathematical models | Diffusion | Convergence | Finite difference method

Numerical approximation | Local limit | Finite difference | Discrete maximum principle | Nonlocal diffusion | Peridynamic models | Convergence analysis | Finite element | peridynamic models | MATHEMATICS, APPLIED | VECTOR CALCULUS | discrete maximum principle | finite element | nonlocal diffusion | BOUNDARY-VALUE-PROBLEMS | ELASTICITY | ANOMALOUS DIFFUSION | convergence analysis | MODELS | SOLID MECHANICS | DYNAMICS | NAVIER EQUATION | CONVERGENCE | finite difference | numerical approximation | local limit | FINITE-ELEMENT-METHOD | Numerical analysis | Approximation | Computer simulation | Mathematical analysis | Mathematical models | Diffusion | Convergence | Finite difference method

Journal Article

Scandinavian Journal of Statistics, ISSN 0303-6898, 03/2018, Volume 45, Issue 1, pp. 194 - 216

We consider fast lattice approximation methods for a solution of a certain stochastic non‐local pseudodifferential operator equation. This equation defines a...

Gaussian Markov random fields | fractional order SPDE | stochastic partial differential equations | inverse problems | spatial interpolation | STATISTICS & PROBABILITY | Approximation | Stochastic processes | Markov processes | Fields (mathematics) | Rounding | Sparse matrices | Continuity (mathematics) | Convergence | Finite difference method

Gaussian Markov random fields | fractional order SPDE | stochastic partial differential equations | inverse problems | spatial interpolation | STATISTICS & PROBABILITY | Approximation | Stochastic processes | Markov processes | Fields (mathematics) | Rounding | Sparse matrices | Continuity (mathematics) | Convergence | Finite difference method

Journal Article

Numerical Algorithms, ISSN 1017-1398, 3/2011, Volume 56, Issue 3, pp. 383 - 403

In this paper, we consider a space-time Riesz–Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion...

Numerical approximation | Short-memory principle | Algorithms | Algebra | Riesz fractional derivative | Stability and convergence | Computer Science | Numeric Computing | Mathematics, general | Theory of Computation | Richardson extrapolation | Caputo fractional derivative | MATHEMATICS, APPLIED | RANDOM-WALKS | PREDICTOR-CORRECTOR APPROACH | FINITE-DIFFERENCE APPROXIMATIONS | Advection-diffusion equation | Extrapolation | Accuracy | Approximation | Mathematical analysis | Mathematical models | Derivatives | Standards

Numerical approximation | Short-memory principle | Algorithms | Algebra | Riesz fractional derivative | Stability and convergence | Computer Science | Numeric Computing | Mathematics, general | Theory of Computation | Richardson extrapolation | Caputo fractional derivative | MATHEMATICS, APPLIED | RANDOM-WALKS | PREDICTOR-CORRECTOR APPROACH | FINITE-DIFFERENCE APPROXIMATIONS | Advection-diffusion equation | Extrapolation | Accuracy | Approximation | Mathematical analysis | Mathematical models | Derivatives | Standards

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 2009, Volume 228, Issue 11, pp. 4038 - 4054

The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative...

Finite differences | Stability | Fractional advection diffusion | DYNAMICS APPROACH | ORDER | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MOTION | STABILITY ANALYSIS | PHYSICS, MATHEMATICAL | DISPERSION EQUATION | ACCURACY

Finite differences | Stability | Fractional advection diffusion | DYNAMICS APPROACH | ORDER | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MOTION | STABILITY ANALYSIS | PHYSICS, MATHEMATICAL | DISPERSION EQUATION | ACCURACY

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 11/2015, Volume 300, pp. 574 - 591

A numerical scheme for the convection–diffusion–reaction (CDR) problems is studied herein. We propose a finite difference method on a special grid for solving...

Finite Element Methods | Finite Difference Methods | Non-uniform grid | Convection–diffusion–reaction | Singular perturbation | Convection-diffusion-reaction | Non uniform grid | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RESIDUAL-FREE BUBBLES | STABILIZATION | STABILITY | ELEMENT METHODS | GALERKIN METHOD | PHYSICS, MATHEMATICAL | Analysis | Algorithms | Bubbles | Approximation | Mathematical analysis | Strategy | Benchmarking | Mathematical models | Diffusion | Finite difference method

Finite Element Methods | Finite Difference Methods | Non-uniform grid | Convection–diffusion–reaction | Singular perturbation | Convection-diffusion-reaction | Non uniform grid | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RESIDUAL-FREE BUBBLES | STABILIZATION | STABILITY | ELEMENT METHODS | GALERKIN METHOD | PHYSICS, MATHEMATICAL | Analysis | Algorithms | Bubbles | Approximation | Mathematical analysis | Strategy | Benchmarking | Mathematical models | Diffusion | Finite difference method

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 05/2019, Volume 77, Issue 9, pp. 2337 - 2353

Radial basis function-generated finite differences (RBF-FD) based on the combination of polyharmonic splines (PHS) with high degree polynomials have recently...

Interpolation | RBF-FD | Polynomials | Runge’s phenomenon | RBF | Meshfree | Runge's phenomenon | MATHEMATICS, APPLIED | FINITE-DIFFERENCES | QUADRATURE | RADIAL BASIS FUNCTIONS | Radial basis function | Approximation | Basis functions | Robustness (mathematics) | Splines | Edge effect | Ill-conditioning (mathematics)

Interpolation | RBF-FD | Polynomials | Runge’s phenomenon | RBF | Meshfree | Runge's phenomenon | MATHEMATICS, APPLIED | FINITE-DIFFERENCES | QUADRATURE | RADIAL BASIS FUNCTIONS | Radial basis function | Approximation | Basis functions | Robustness (mathematics) | Splines | Edge effect | Ill-conditioning (mathematics)

Journal Article

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 07/2018, Volume 115, Issue 4, pp. 462 - 500

Summary In this work, we are concerned with radial basis function–generated finite difference (RBF‐FD) approximations. Numerical error estimates are presented...

structured point clouds | supplementary polynomials | radial basis function–generated finite difference | stabilized Gaussians | polyharmonic splines | unstructured point clouds | INCOMPRESSIBLE VISCOUS FLOWS | SIMULATION | DIFFERENTIAL QUADRATURE METHOD | INTERPOLATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | NAVIER-STOKES EQUATIONS | ENGINEERING, MULTIDISCIPLINARY | FINITE-DIFFERENCES | radial basis function-generated finite difference | INTERFACES | RADIAL BASIS FUNCTIONS | COMPUTATION | SHAPE PARAMETER | Radial basis function | Poisson equation | Polynomials | Basis functions | Splines

structured point clouds | supplementary polynomials | radial basis function–generated finite difference | stabilized Gaussians | polyharmonic splines | unstructured point clouds | INCOMPRESSIBLE VISCOUS FLOWS | SIMULATION | DIFFERENTIAL QUADRATURE METHOD | INTERPOLATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | NAVIER-STOKES EQUATIONS | ENGINEERING, MULTIDISCIPLINARY | FINITE-DIFFERENCES | radial basis function-generated finite difference | INTERFACES | RADIAL BASIS FUNCTIONS | COMPUTATION | SHAPE PARAMETER | Radial basis function | Poisson equation | Polynomials | Basis functions | Splines

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 11/2015, Volume 300, pp. 695 - 709

We construct accurate central difference stencils for problems involving high frequency waves or multi-frequency solutions over long time intervals with a...

Approximation theory | Wave propagation | Wavenumber approximation | Dispersion relation | Finite differences | HIGH-ACCURACY | ORDER | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | GROUP-VELOCITY | PHYSICS, MATHEMATICAL | PROPAGATION | SCHEMES | Errors | Construction | Approximation | Infinity | Mathematical analysis | Norms | Mathematical models | High frequencies | Naturvetenskap | Computational Mathematics | Mathematics | Natural Sciences | Beräkningsmatematik | Matematik | Dispersion relation; Wave propagation; Wavenumber approximation; Finite differences; Approximation theory

Approximation theory | Wave propagation | Wavenumber approximation | Dispersion relation | Finite differences | HIGH-ACCURACY | ORDER | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | GROUP-VELOCITY | PHYSICS, MATHEMATICAL | PROPAGATION | SCHEMES | Errors | Construction | Approximation | Infinity | Mathematical analysis | Norms | Mathematical models | High frequencies | Naturvetenskap | Computational Mathematics | Mathematics | Natural Sciences | Beräkningsmatematik | Matematik | Dispersion relation; Wave propagation; Wavenumber approximation; Finite differences; Approximation theory

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 2011, Volume 62, Issue 3, pp. 938 - 944

A one-dimensional fractional diffusion model is considered, where the usual second order derivative gives place to a fractional derivative of order α , with 1...

Fractional diffusion | Integro-differential equations | Finite differences | MATHEMATICS, APPLIED | TRANSPORT | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | ADVECTION-DISPERSION EQUATION | STABILITY | FINITE-DIFFERENCE APPROXIMATIONS | Approximation | Computer simulation | Mathematical analysis | Splines | Consistency | Mathematical models | Derivatives | Diffusion

Fractional diffusion | Integro-differential equations | Finite differences | MATHEMATICS, APPLIED | TRANSPORT | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | ADVECTION-DISPERSION EQUATION | STABILITY | FINITE-DIFFERENCE APPROXIMATIONS | Approximation | Computer simulation | Mathematical analysis | Splines | Consistency | Mathematical models | Derivatives | Diffusion

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 03/2019, Volume 77, Issue 6, pp. 1770 - 1785

Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition,...

Homogenization | Photonic devices | Finite difference schemes | Trefftz approximations | Maxwell’s equations | Wave scattering | Maxwell's equations | MATHEMATICS, APPLIED | DECOMPOSITION | SINGULAR-VALUE | INTERPOLATION | HELMHOLTZ-EQUATION | DIFFERENCE | DISCONTINUOUS GALERKIN METHODS | ULTRA-WEAK | COMPUTATION | FINITE-ELEMENT-METHOD | Electromagnetic radiation | Numerical analysis | Electromagnetism | Electromagnetic waves | Differential equations | Electric waves | Interpolation | Accuracy | Boundary value problems | Computer simulation | Mathematical analysis | Complex media | Well posed problems | Matrix methods | Periodic structures | Photonics

Homogenization | Photonic devices | Finite difference schemes | Trefftz approximations | Maxwell’s equations | Wave scattering | Maxwell's equations | MATHEMATICS, APPLIED | DECOMPOSITION | SINGULAR-VALUE | INTERPOLATION | HELMHOLTZ-EQUATION | DIFFERENCE | DISCONTINUOUS GALERKIN METHODS | ULTRA-WEAK | COMPUTATION | FINITE-ELEMENT-METHOD | Electromagnetic radiation | Numerical analysis | Electromagnetism | Electromagnetic waves | Differential equations | Electric waves | Interpolation | Accuracy | Boundary value problems | Computer simulation | Mathematical analysis | Complex media | Well posed problems | Matrix methods | Periodic structures | Photonics

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 2006, Volume 56, Issue 1, pp. 80 - 90

Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in...

Two-sided fractional partial differential equation | Fractional PDE | Finite difference approximation | Stability | Left-handed fractional flow | Implicit Euler method | Right-handed fractional flow | Backward Euler method | Numerical fractional PDE | Fractional derivative | MATHEMATICS, APPLIED | backward Euler method | NUMERICAL-SOLUTION | two-sided fractional partial differential equation | finite difference approximation | fractional PDE | implicit Euler method | left-handed fictional flown | right-handed fractional flow | fractional derivative | numerical fractional PDE | stability

Two-sided fractional partial differential equation | Fractional PDE | Finite difference approximation | Stability | Left-handed fractional flow | Implicit Euler method | Right-handed fractional flow | Backward Euler method | Numerical fractional PDE | Fractional derivative | MATHEMATICS, APPLIED | backward Euler method | NUMERICAL-SOLUTION | two-sided fractional partial differential equation | finite difference approximation | fractional PDE | implicit Euler method | left-handed fictional flown | right-handed fractional flow | fractional derivative | numerical fractional PDE | stability

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 07/2019, Volume 351, pp. 184 - 225

In this work, we study the finite difference approximation for a class of nonlocal fracture models. The nonlocal model is initially elastic but beyond a...

Finite difference approximation | Numerical analysis | Nonlocal fracture models | State based peridynamics | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | DYNAMIC CRACK-PROPAGATION | ENGINEERING, MULTIDISCIPLINARY | LIMIT | NONLOCAL DIFFUSION | HORIZON | Analysis | Models | Finite element method | Tensile strain | Computer simulation | Mathematical analysis | Approximations | Mathematical models | Numerical prediction | Well posed problems | Continuity (mathematics) | Convergence | Crack propagation | Finite difference method

Finite difference approximation | Numerical analysis | Nonlocal fracture models | State based peridynamics | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | DYNAMIC CRACK-PROPAGATION | ENGINEERING, MULTIDISCIPLINARY | LIMIT | NONLOCAL DIFFUSION | HORIZON | Analysis | Models | Finite element method | Tensile strain | Computer simulation | Mathematical analysis | Approximations | Mathematical models | Numerical prediction | Well posed problems | Continuity (mathematics) | Convergence | Crack propagation | Finite difference method

Journal Article