Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

Abstract: We establish uniform error estimates of finite difference methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter $\varepsilon$ ($\varepsilon\in(0,1]$). When $\varepsilon\to0^+$, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., $0<\varepsilon\ll1$, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with $O(\varepsilon^2)$-wavelength at $O(\varepsilon^4)$ and $O(\varepsilon^2)$ amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in $\varepsilon$ of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in $\varepsilon$, at the order of $O(h^2+\tau)$ and $O(h^2+\tau^{2/3})$ with time step $\tau$ and mesh size $h$ for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the $l^2$-norm and discrete semi-$H^1$ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine $\varepsilon$-dependent error estimates of NLSW, $\varepsilon$-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-$H^1$ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.

Propagation in linear dispersive media: finite difference time-domain methodologies

Abstract: Finite difference time-domain (FDTD) methodologies are presented for electromagnetic wave propagation in two different kinds of linear dispersive media: an Nth order Lorentz and an Mth order Debye medium. The temporal discretization is accomplished by invoking the central difference approximation for the temporal derivatives that appear in the first-order differential equations. From this, the final equations are temporally advanced using the classical leapfrog method. One-dimensional scattering from a dielectric slab is chosen for a test case. Provided that the maximum operating frequency times the time step is small and that the wave is adequately resolved in space, as shown in the error analysis, the agreement between the computed and exact solutions will be excellent. The attached data, which are associated with the four pole Lorentz dielectric and the five pole Debye medium, verify this assertion. >

On the finite difference solution of two-dimensional induction problems

Abstract: Summary The numerical solution by finite differences of two-dimensional problems in electromagnetic induction is reexamined with a view to generalizing the method to three-dimensional models. Previously published work, in which fictitious values were used to derive the finite difference equations, is discussed and some errors in the theory which appear to have gone undetected so far, are pointed out. It is shown that the previously published B-polarization formulas are incorrect at points where regions of different conductivity meet, and that the E-polarization formulas are inaccurate when the step sizes of the numerical grid around the point are uneven. An appropriately-modified version of the two-dimensional theory is developed on the assumption that the Earth's conductivity is a smoothly-varying function of position, a method which naturally lends itself to three-dimensional generalization. All the required finite-difference formulas are derived in detail, and presented in a form which is suitable for programming. A simple numerical calculation is given to illustrate the application of the method and the results are compared with those obtained from previous work.

Wave and Scattering Methods for Numerical Simulation

Abstract: Preface.Foreword.1. Introduction.1.1 An Overview of Scattering Methods.1.1.1 Remarks on Passivity.1.1.2 Case Study: The Kelly-Lochbaum Digital Speech Synthesis Mode.1.1.3 Digital Waveguide Networks.1.1.4 A General Approach: Multidimensional Circuit Representations and Wave Digital Filters.1.2 Questions.2. Wave Digital Filters.2.1 Classical Network Theory.2.1.1 N-ports.2.1.2 Power and Passivity.2.1.3 Kirchhoff's Laws.2.1.4 Circuit Elements.2.2 Wave Digital Elements and Connections.2.2.1 The Bilinear Transform.2.2.2 Wave Variables.2.2.3 Pseudopower and Pseudopassivity.2.2.4 Wave Digital Elements.2.2.5 Adaptors.2.2.6 Signal and Coefficient Quantization.2.2.7 VectorWave Variables.2.3 Wave Digital Filters and Finite Differences.3. Multidimensional Wave Digital Filters.3.1 Symmetric Hyperbolic Systems.3.2 Coordinate Changes and Grid Generation.3.2.1 Structure of Coordinate Changes.3.2.2 Coordinate Changes in (1 +1)D.3.2.3 Coordinate Changes in Higher Dimensions.3.3 MD-passivity.3.4 MD Circuit Elements.3.4.1 The MD Inductor.3.4.2 OtherMD Elements.3.4.3 Discretization in the Spectral Domain.3.4.4 Other Spectral Mappings.3.5 The (1 +1)D Advection Equation.3.5.1 A Multidimensional Kirchhoff Circuit.3.5.2 Stability.3.5.3 An Upwind Form.3.6 The (1 +1)D Transmission Line.3.6.1 MDKC for the (1 + 1)D Transmission Line Equations.3.6.2 Digression: The Inductive Lattice Two-port.3.6.3 Energetic Interpretation.3.6.4 A MDWD Network for the (1 + 1)D Transmission Line.3.6.5 Simplified Networks.3.7 The (2 +1)D Parallel-plate System.3.7.1 MDKC and MDWD Network.3.8 Finite-difference Interpretation.3.8.1 MDWD Networks as Multistep Schemes.3.8.2 Numerical Phase Velocity and Parasitic Modes.3.9 Initial Conditions.3.10 Boundary Conditions.3.10.1 MDKC Modeling of Boundaries.3.11 Balanced Forms.3.12 Higher-order Accuracy.4. Digital Waveguide Networks.4.1 FDTD and TLM.4.2 Digital Waveguides.4.2.1 The Bidirectional Delay Line.4.2.2 Impedance.4.2.3 Wave Equation Interpretation.4.2.4 Note on the Different Definitions of Wave Quantities.4.2.5 Scattering Junctions.4.2.6 Vector Waveguides and Scattering Junctions.4.2.7 Transitional Note.4.3 The (1 +1)D Transmission Line.4.3.1 First-order System and the Wave Equation .4.3.2 Centered Difference Schemes and Grid Decimation.4.3.3 A (1+1)D Waveguide Network.4.3.4 Waveguide Network and the Wave Equation.4.3.5 An Interleaved Waveguide Network.4.3.6 Varying Coefficients.4.3.7 Incorporating Losses and Sources.4.3.8 Numerical Phase Velocity and Dispersion.4.3.9 Boundary Conditions.4.4 The (2 +1)D Parallel-plate System.4.4.1 Defining Equations and Centered Differences.4.4.2 The Waveguide Mesh.4.4.3 Reduced Computational Complexity and Memory Requirements in the Standard Form of the Waveguide Mesh.4.4.4 Boundary Conditions.4.5 Initial Conditions.4.6 Music and Audio Applications of Digital Waveguides.5. Extensions of Digital Waveguide Networks.5.1 Alternative Grids in (2 +1)D.5.1.1 Hexagonal and Triangular Grids.5.1.2 The Waveguide Mesh in Radial Coordinates.5.2 The (3 + 1)D Wave Equation and Waveguide Meshes.5.3 The Waveguide Mesh in General Curvilinear Coordinates.5.4 Interfaces between Grids.5.4.1 Doubled Grid Density Across an Interface.5.4.2 Progressive Grid Density Doubling.5.4.3 Grid Density Quadrupling.5.4.4 Connecting Rectilinear and Radial Grids.5.4.5 Grid Density Doubling in (3 +1)D.5.4.6 Note.6. Incorporating the DWN into the MDWD Framework.6.1 The (1 +1)D Transmission Line Revisited.6.1.1 Multidimensional Unit Elements.6.1.2 Hybrid Form of the Multidimensional Unit Element.6.1.3 Alternative MDKC for the (1+1)D Transmission Line.6.2 Alternative MDKC for the (2 + 1)D Parallel-plate System.6.3 Higher-order Accuracy Revisited.6.4 Maxwell's Equations.7. Applications to Vibrating Systems.7.1 Beam Dynamics.7.1.1 MDKC and MDWDF for Timoshenko's System.7.1.2 Waveguide Network for Timoshenko's System.7.1.3 Boundary Conditions in the DWN.7.1.4 Simulation: Timoshenko's System for Beams of Uniform and Varying Cross-sectional Areas.7.1.5 Improved MDKC for Timoshenko's System via Balancing.7.2 Plates.7.2.1 MDKCs and Scattering Networks for Mindlin's System.7.2.2 Boundary Termination of the Mindlin Plate.7.2.3 Simulation: Mindlin's System for Plates of Uniform and Varying Thickness.7.3 Cylindrical Shells.7.3.1 The Membrane Shell.7.3.2 The Naghdi-Cooper System II Formulation.7.4 Elastic Solids.7.4.1 Scattering Networks for the Navier System.7.4.2 Boundary Conditions.8. Time-varying and Nonlinear Systems.8.1 Time-varying and Nonlinear Circuit Elements.8.1.1 Lumped Elements.8.1.2 Distributed Elements.8.2 Linear Time-varying Distributed Systems.8.2.1 A Time-varying Transmission Line Model.8.3 Lumped Nonlinear Systems in Musical Acoustics.8.3.1 Piano Hammers.8.3.2 The Single Reed.8.4 From Wave Digital Principles to Relativity Theory.8.4.1 Origin of the Challenge.8.4.2 The Principle of Newtonian Limit.8.4.3 Newton's Second Law.8.4.4 Newton's Third Law and Some Consequences.8.4.5 Moving Electromagnetic Field.8.4.6 The Bertozzi Experiment.8.5 Burger's Equation.8.6 The Gas Dynamics Equations.8.6.1 MDKC and MDWDF for the Gas Dynamics Equations.8.6.2 An Alternate MDKC and Scattering Network.8.6.3 Entropy Variables.9. Concluding Remarks.9.1 Answers.9.2 Questions.A. Finite Difference Schemes for the Wave Equation.A.1 Von Neumann Analysis of Difference Schemes.A.1.1 One-step Schemes.A.1.2 Multistep Schemes.A.1.3 Vector Schemes.A.1.4 Numerical Phase Velocity.A.2 Finite Difference Schemes for the (2 + 1)D Wave Equation.A.2.1 The Rectilinear Scheme.A.2.2 The Interpolated Rectilinear Scheme.A.2.3 The Triangular Scheme.A.2.4 The Hexagonal Scheme.A.2.5 Note on Higher-order Accuracy.A.3 Finite Difference Schemes for the (3 + 1)D Wave Equation.A.3.1 The Cubic Rectilinear Scheme.A.3.2 The Octahedral Scheme.A.3.3 The (3 + 1)D Interpolated Rectilinear Scheme.A.3.4 The Tetrahedral Scheme.B. Eigenvalue and Steady State Problems.B.1 Introduction.B.2 Abstract Time Domain Models.B.3 Typical Eigenvalue Distribution of a Discretized PDE.B.4 Excitation and Filtering.B.5 Partial Similarity Transform.B.6 Steady State Problems.B.7 Generalization to Multiple Eigenvalues.B.8 Numerical Example.Bibliography.Index.