Stability of leapfrog method. Recall from earlier … .

Stability of leapfrog method An analysis found in IEI shows that the solution is now divergent for all real numerical values of . However, it limits the choice of the time Jun 2, 2022 · In section2, we give a brief introduction of leapfrog and Crank-Nicolson FDTD. We also show that Gragg’s smoothing scheme Aug 28, 2017 · Recap. Also presented are a stability analysis, an accuracy analysis based on the Jun 1, 2023 · The stability of the leapfrog ADI-FDTD method is analyzed using the Fourier method and the eigenvalues of the Fourier amplification matrix are obtained analytically to prove the Apr 5, 2022 · Abstract. –, are lagged at t = t n − Δt in the We propose and analyze a linear stabilization of the Crank-Nicolson Leapfrog (CNLF) method that removes all time step/CFL conditions for stability and controls the unstable mode. Introduction. Despite the mentioned attractive properties, the method has Jun 5, 2022 · 1 Background The N-Body Problem considers nbodies in R3 which interact solely via gravitational forces. from publication: On the Nonlinear Stability of Symplectic Integrators | The modified Hamiltonian is used to study the Nov 16, 2015 · The aim of this section is to derive our computational method. 2) u n + 1 = u n − 1 + 2 Δ t f (u n), where Δt denotes the fixed timestep, Abstract The leapfrog method is popular because of its good stability when solving partial differential equations with oscillatory solutions. Sep 14, 2022 · Conditional stability, IMEX methods, Crank-Nicolson, Leap-Frog, Robert-Asselin filter AMS subject classifications. It is obtained by natural modifications of Jun 30, 2016 · The Leapfrog method for the solution of Ordinary Differential Equation initial value problems has been historically popular for several reasons. 1 Stability Analysis of Leapfrog To analyse the stability of a time-stepping scheme for solving a wave or advection equation, we analyse how the scheme behaves for the 1D Sep 5, 2023 · We also show that Gragg's smoothing scheme improves the stability of the method. 20, 2014 Context: consider the initial value problem for linear time-dependent PDEs. 1016/j. Its unconditional stability is analytically proven and This report proves that under the time step condition ∆t|Λ| < 1 (| · | = Euclidean norm) suggested by root condition analysis and necessary for stability, all modes of the CrankNicolson Leap Dec 4, 2024 · Convergence and stability of a high-order leap-frog based discontinuous Galerkin method for the Maxwell equations on non-conforming meshes Hassan Fahs, Stephane Lanteri Dec 9, 2014 · We propose and analyze a linear stabilization of the Crank-Nicolson Leapfrog (CNLF) method that removes all time step/CFL conditions for stability and controls the Jan 10, 2023 · We study the probability and energy conservation properties of a leap-frog finite-difference time-domain (FDTD) method for solving the Schrödinger equation. Problem setting 1 October 17 Andreas Jul 21, 2016 · In this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. The diffusion terms, which are the last terms in Eqs. Courant-Friedrichs-Lewy (CFL) Stability Criterion Let’s consider the stability condition obtained above using the concept of domain of dependence. The CNLF method was used to solve the nonstationary Stokes-Darcy or Navier Jul 28, 2016 · The Leapfrog method is efficient as it only requires one function evaluation per time step, with a function evaluation referring to the function F in Equation (1). Two ways to derive Jan 1, 2012 · The stability of the leapfrog ADI-FDTD method is analyzed using the Fourier method and the eigenvalues of the Fourier amplification matrix are obtained analytically to prove the Stability of a Leap-Frog Discontinuous Galerkin Method for Time-Domain Maxwell's Equations in Anisotropic Materials - Volume 21 Issue 5. Modified 8 years, 3 months ago. Furthermore, the dispersion relation of the leapfrog ADI-FDTD method is also not clear. $$ Feb 1, 2018 · Usually, the standard leap-frog scheme is accompanied by other application of finite difference schemes to overcome certain unstable issues. 025. “It has the disadvantage that the solution at odd time steps Stability regions in the complex Jδt plane of the filtered leapfrog, the third-order Adams-Bashforth, and the fourth-order Runge-Kutta schemes, all adjusted to possess the same average frequency Nov 24, 2017 · Kong Yongdan, Chu Qingxin. 300 MIT, (Rosales) Notes: vNSA von Neumann Stability Analysis. • von Neumann Stability: Nov 2, 2008 · • Consistency: It can be shown (exercise) that the method is consistent if and only if P N(1) = 0 and P N (1) = Q N(1). The second-order leap-frog predictor is given by Sep 14, 2022 · We focus on the accuracy and the stability of the leapfrog scheme combined with the Robert{Asselin{Williams lter, the higher-order Robert{Asselin type time lter, the composite- Implementation of the leapfrog ADI-FDTD method for lossy media with special consideration for boundary con- dition is provided. Download scientific diagram | 5: A stable orbit for the leapfrog method. 3 The Wave Equation and Staggered Leapfrog This section focuses on the second-order wave equation utt = c2uxx. Thus, Euler-Cromer becomes leapfrog simply by updating the velocity by an extra half-step at the beginning, and using the resulting value of v as the starting value in Eq. We will consider a nodal discontinuous Galerkin method for the space discretization and a leap-frog method for A one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method including lumped elements is presented and its unconditional stability is analytically proven by DOI: 10. 1. “It has the disadvantage that the solution at odd DOI: 10. It finds that while the method is stable for Shared from Wolfram Cloud In this communication, higher-order convolutional perfectly matched layer (HO-CPML) boundary conditions have been developed for the leapfrog weakly conditional stability finite-difference Jul 2, 2020 · e. Then, an alternating Apr 21, 2014 · times. The latter O ( h 2 ) framework enjoys more Mar 16, 2024 · The document summarizes research on the stability of the leapfrog/midpoint method for solving differential equations numerically. The fundamental Feb 24, 2025 · the exact solution u(x, t). However, we require that the numerical dependency cone is Oct 3, 2018 · term stability and freedom from unphysical damping are essential. Viewed 765 times Mar 4, 2023 · the stability of the leapfrog ADI-FDTD method. The Feb 9, 2019 · $\begingroup$ As with every multi-step method you need one or multiple starter steps that are provided by a different method. Furthermore the artificial Generalized Leapfrog Methods A. This seems to be a first proof of Oct 1, 2019 · Single step explicit and implicit methods and the multi-step central difference scheme, also known as the leapfrog or the midpoint method, have been applied to these Stability analysis of the Crank–Nicolson-Leapfrog method with the Robert–Asselin–Williams time filter Mar 8, 2022 · The non-empty regions of these two methods and the regions of other methods based on the same definition seem more informative in explaining the behavior of those Convergence and stability of a high-order leap-frog based discontinuous Galerkin method for the Maxwell equations on non-conforming meshes The Robert-Asselin (RA) time filter combined with leapfrog scheme is widely used in numerical models of weather and climate. Accuracy and Feb 1, 2009 · Semantic Scholar extracted view of "Stability of the leapfrog/midpoint method" by L. 09. We nd the exact solution u(x;t). Jia et al. g. Numerical resulted to presented to gauge the accuracy and stability of the proposed algorithm. The leapfrog CDI-FDTD method is May 10, 2023 · This paper presents decoupled second-order accurate algorithm based on Crank–Nicolson LeapFrog (CNLF) scheme for the evolution Boussinesq equations. Convergence of an explicit iterative leap-frog Jan 30, 2024 · The implicit-explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Geophysical flow simulations have evolved sophisticated implicit–explicit time Jul 4, 2014 · Numerical solution of partial differential equations governing time domain simulations in computational electromagnetics, is usually based on grid methods in space and on explicit Stability of a Leap-Frog Discontinuous Galerkin Method for Time-Domain Maxwell's Equations in Anisotropic Materials TY - JOUR T1 - Stability of a Leap-Frog Discontinuous Galerkin Jun 1, 1974 · For smooth problems where dissipation is not wanted the Leapfrog method seems to be an ideal method. The source term is eliminated by transforming the May 31, 2024 · We propose and analyze an efficient, unconditionally stable, second order convergent, artificial compressibility Crank--Nicolson leap-frog (CNLFAC) method for Jul 22, 2024 · A graphical description of the Leapfrog Method# The numerical leap-frog scheme recreates this dependency cone. This numerical flaw is a drawback not present in the conventional time-collocated ADI-FDTD method and constitutes a Nov 13, 2023 · An Introduction to Multistep Methods: Leap-frog# References: Section 6. The attention will be restricted to second order methods for May 1, 2005 · Abstract The stability of constant-coefficients semi-implicit schemes for the hydrostatic primitive equations and the fully elastic Euler equations in the presence of explicitly Nov 1, 2023 · The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the May 5, 2023 · The preceding works introduced the leapfrog complying divergence implicit finite-difference time-domain (CDI-FDTD) method, which exhibits high accuracy and unconditional The LADI-FDTD (leapfrog alternative direction implicit finite difference time domain method), was widely used to solve the problem of electromagnetic calcula-tion on large size because it Mar 11, 2021 · The stability of the leapfrog ADI-FDTD method is analyzed using the Fourier method and the eigenvalues of the Fourier amplification matrix are obtained analytically to Abstract The instability encountered by applying the upwind leapfrog method to the advection equation having a source term is resolved. In other words, stability means kSNk ≤ C. Shampine. 4) Below we give some examples of IMEX multistep methods with the stability regions of the explicit methods. Efficient unconditionally stable one step leapfrog ADI-FDTD method with low numerical dispersion[J]. However, it limits the choice of the As usual for explicit time discretisation schemes, the numerical stability is Feb 1, 2012 · For CNLF we prove stability for the coupled system under the time step condition suggested by linear stability theory for the Leap-Frog scheme. 1016/0021-9991(84)90039-1 Corpus ID: 120042888; Analytical, linear stability criteria for the leap-frog, Dufort-Frankel method @article A leapfrog scheme for the unconditionally stable complying-divergence implicit (CDI) finite-difference time-domain (FDTD) method is presented. Comparisons of the efficiency, stability, convergence rate for our modified method Mar 6, 2000 · 2. In this paper, we study the stability of the Crank–Nicolson-Leapfrog scheme with the May 27, 2020 · Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. 1 Stability Analysis of Leapfrog To analyse the stability of a time-stepping scheme for solving a wave or advection equation, we analyse how the scheme behaves for the 1D The leapfrog method is popular because of its good stability when solving partial differential equations with oscillatory solutions. 1 Further considerations regarding stability The restriction jGj 1 for all in (1. In this paper, we present the rigorous analysis of the Jan 1, 2022 · The stability of the leapfrog ADI-FDTD method is analyzed using the Fourier method and the eigenvalues of the Fourier amplification matrix are obtained analytically to prove the Aug 1, 2024 · Recently proposed leapfrog complying-divergence implicit-finite-difference time-domain (CDI-FDTD) method features many advantages over other unconditionally stable Semantic Scholar extracted view of "Decoupled Stabilized Crank-Nicolson LeapFrog method for time-dependent Navier-Stokes/Darcy model" by X. GROTE, Abstract Geophysical flow simulations have evolved sophisticated implicit–explicit time stepping methods (based on fast-slow wave splittings) followed by time filters to control any unstable Feb 1, 1984 · The criterion of linear numerical stability of the combined leap-frog Dufort-Frankel scheme for advective-diffusive problems in two dimensions is κ x Δx 2 + κ y Δy 2 U 2 κ x + V 2 Apr 21, 2014 · times. Its stability under a CFL condition Jun 2, 2013 · A one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is proposed for modelling anisotropic magnetized plasma. 1) u ′ (t) = f (u (t)), u (0) = u 0 is given by (1. Difierent from the method provided by others, the Oct 1, 2019 · Compared to the single-step implicit scheme however, the super-stability is more pronounced in the midpoint (leapfrog) method. This two-step method requires that we Jan 27, 2025 · Analysing the numerical stability of leap-frog method for different timesteps. The eigenvalues of amplification matrix are obtained to prove the Jul 22, 2016 · In this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. We Sep 1, 2018 · The leapfrog (LF) scheme applied to the initial value problem (1. Usually, the long-time instability of Apr 27, 2022 · 18. We present a sufficient Sep 14, 2022 · The implicit-explicit combination of Crank-Nicolson and Leap-Frog methods is widely used for atmo-sphere, ocean and climate simulations. However, the leapfrog ADI-FDTD method needs to solve six implicit equations in one Oct 1, 2020 · We establish the unconditional long-time stability of the scheme. The stability analysis shows Nov 26, 2013 · A second order explicit one-step numerical method for the initial value problem of the general ordinary differential equation is proposed. 7 Multistep Methods of [Sauer, 2019]. We emphasize Mar 1, 2015 · In this paper, at first, the stability condition is addressed in a general way by allowing the time step increment get away from the minimum points spacing. Although it was a good idea to center both A one-step leapfrog alternating-direction-implicit flnite- difierence time-domain (ADI-FDTD) method for lossy media is presented. By combination of Crank–Nicolson Jul 1, 2014 · The unfiltered leapfrog method and the LF-MMK use a larger time step of Δt = 0. • Note that the Jun 1, 2015 · We propose and analyze a linear stabilization of the Crank–Nicolson Leapfrog (CNLF) method that removes all time step/CFL conditions for stability and controls the Mar 19, 2020 · Backwards Euler Method/ Implicit Euler Scheme - Difference Equation + Stability Hot Network Questions Would an East-Ender in London in Victorian England have understood Jan 8, 2015 · 1 Stability of the leapfrog scheme 2 The phase shift of the leapfrog scheme 3 The Lax-Wendroff scheme 4 LTE, stability, and phase shift of the Lax-Wendroff scheme M. Geophysical flow simulations have evolved Jun 30, 2015 · We propose and analyze a linear stabilization of the Crank–Nicolson Leapfrog (CNLF) method that removes all time step/CFL conditions for stability and controls the The leap-frog DG method in anisotropic materials which is discussed in [4] leads to a locally implicit method for the case of SM-ABC. 76D05, 65L20, 65M12 1. Semantic May 27, 2014 · The fundamental method for time stepping in most current geophysical fluid dynamics (GFD) codes consists of one step of the Crank–Nicolson-Leapfrog (CNLF) method Feb 24, 2014 · Stability and Leapfrog Scheme MIT 18. The method has second order Apr 11, 2022 · I try to understand the Fourier stability analysis for Leap-Frog scheme to solve linear advection equation, $$\dfrac{\partial u}{\partial t}+a\dfrac{\partial u}{\partial x}=0. In [1] a fully explicit in time leap-frog DG method is Oct 30, 2024 · The method has second order accuracy, requires only one function evaluation per time step, and is non-dissipative. We illustrate two applications of the method: uncoupling IMS method which has excellent stability and accuracy properties for the ODEs arising from the Euler equations is the iterated leap-frog method. Where boundaries do not cause any complications, e. “It has the disadvantage that the solution at odd Jun 6, 2016 · 1. 026 Corpus ID: 9013452; A Crank-Nicolson Leapfrog stabilization: Unconditional stability and two applications @article{Jiang2015ACL, title={A Crank-Nicolson In this work we discuss the numerical discretization of the time-dependent Maxwell’s equations using a fully explicit leap-frog type discontinuous Galerkin method. $$ As both characteristic roots Oct 8, 2013 · The staggered leapfrog method simply uses a centered scheme for the time derivative; so, for instance, for the advection we would get, If we perform a von Neumann Apr 23, 2014 · by numerical approximation using the Leapfrog Method Linear Multistep Method: Leapfrog •A general form for s-step linear multi-step methods will have the form ∑s m=0 To analyze the stability of the leapfrog ADI-FDTD method for lossy media, the von Neumann method combining with the Jury criterion is employed [12]. 086 Feb. Skip to search form Skip to main content Skip to account menu. 10) is a bit of an over Aug 15, 2019 · popular explicit leapfrog method for the wave equation. The source term is eliminated by Stability and dispersion analysis for the three-dimensional (3-D) leapfrog alternate direction implicit flnite difierence time domain (ADI-FDTD) method is presented and the eigenvalues of Jan 9, 2016 · All you need to do is find the stability criteria for whatever scheme (in this case Leap Frog Method) with respect to the simple model equation: $$\frac{dz}{dt} = \lambda z$$ for Feb 25, 2017 · The Courant–Friedrichs–Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. Apr 10, 2021 · $\begingroup$ @mattos : This is the central Euler method or 2-step Nystrom method, with the cited weak stability. Try the Euler methods or the Crank-Nicolson Jan 23, 2023 · Wave propagation, finite element methods, explicit time integration, leap-frog method, convergence theory, damped Chebyshev polynomials. We will confine ourselves Feb 24, 2014 · • Stability: kSNUk ≤ CkUk where C is a constant independent of ∆t. global Jul 20, 2003 · The instability encountered by applying the upwind leapfrog method to the advection equation having a source term is resolved. However, to solve the fully discrete nonlinear systems, these methods generally require the A conformal leapfrog alternating direction implicit finite-difference time-domain (CLeapfrog ADI-FDTD) method based on conformal technique was proposed in the article. Here N = T ∆t is the total time steps. Denote the fully discretized scheme as Nov 2, 2008 · • This method was not needed here because we could use the quadratic formula. 3. edu. of Discretization Methods There are several distinct approaches to the formulation of computer methods for solving differential equations. Numerical results and examples are presented to validate Aug 28, 2017 · effect on stability of the overall process has been tested in numerous simulations. A solution to the N-Body Problem will be the derivation of a set of Jun 6, 2016 · 1. Contributed by: Ulrich Mutze (2011) Open content The stability of the Crank–Nicolson-Leapfrog scheme with the Robert–Asselin–Williams time filter is studied. 2014. It successfully suppresses the spurious computational mode Jan 1, 2016 · The Crank-Nicolson-Leap Frog (CNLF) method is a second-order scheme that employs the implicit Crank-Nicolson discretization of subdomain terms and treats the interface May 5, 2020 · Free from the Courant-Friedrichs-Lewy (CFL) stability condition and sub-step computations, the one-step leap-frog alternative-direction-implicit finite-difference time-domain May 1, 2020 · The eigenvalues of amplification matrix are obtained to prove the unconditional stability. 23) by interpolation method. However, for higher order methods this technique becomes extremely useful. Compared with the Oct 1, 2017 · The CNLF method was first analyzed in [15], and stability for the CNLF method was proven in [16]. In section3, we provide an overview of the general leapfrog time-stepping equation of [19,2], and May 27, 2015 · Here the result is even worse. ISERLES King's College, University of Cambridge, Cambridge CB2 1ST, England The stability of such methods is analysed by using the order-star theory. It also Jul 1, 2024 · The main characteristics of the leap-frog scheme are the second-order convergence scheme, which maintains wave conservation and solves certain instability problems, attracting Jan 16, 2021 · In this paper we propose and analyze an unconditionally stable leapfrog method for Maxwell’s equations that removes the time step constraint for stability, which makes the Jan 1, 2013 · The stability of the leapfrog ADI-FDTD method is analyzed using the Fourier method and the eigenvalues of the Fourier amplification matrix are obtained analytically to prove the An one-step arbitrary-order leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is presented. kit. We present a Dec 1, 2014 · Stability is proven for two second order, two step methods for uncoupling a system of two evolution equations with exactly skew symmetric coupling: the Crank-Nicolson Leap The Leapfrog method is efficient as it only requires one function evaluation per time step, with a function The stability of ODE methods is a relatively new concept in numerical methods and Jun 17, 2003 · The instability encountered by applying the upwind leapfrog method to the advection equation having a source term is resolved. We propose May 27, 2014 · The stability of the Crank–Nicolson-Leapfrog scheme with the Robert–Asselin–Williams time filter is studied. The leapfrog scheme Jun 28, 2024 · leapfrog ADI-FDTD method for modeling lossy media. Accuracy and stability are confirmed for the leapfrog method (centered second differences in t and x). 2 MARCUS J. 4 1. , the second-order Adams-Bashforth method [18], the third-order Adam-Bashforth [8], the leapfrog-trapezoidal method [14, 33] or the Magazenkov method [19]. • 0-stability: The method is said to be 0-stable provided the Feb 1, 2009 · The leapfrog method is popular because of its good stability when solving partial differential equations with oscillatory solutions. $$ The region of absolute stability is Aug 1, 2017 · Although the original Alternating-Direction-Implicit (ADI) FDTD method was a split-step method where additional field variables were needed at intermediate time instances, it Sep 1, 2018 · The RA and RAW filters successfully suppress the spurious computational mode associated with the leapfrog method, but also weakly damp the physical mode and degrade Request PDF | Stability of a leap-frog discontinuous Galerkin method for time-domain Maxwell's equations in anisotropic materials | In this work we discuss the numerical discretization of the Mar 11, 2021 · Using the Fourier (von Neumann) approach, the stability of the leapfrog CDI-FDTD method is analyzed. The ubiquitous May 22, 2019 · The implicit-explicit combination of Crank-Nicolson and Leap-Frog methods is widely used for atmosphere, ocean and climate simulations. Jan 1, 2008 · Convergence and stability of a high-order leap-frog based discontinuous Galerkin method for the Maxwell equations on non-conforming meshes Jun 1, 2023 · The convex splitting method is unconditionally energy-stable and uniquely solvable. cam. Furthermore, the leapfrog alternating direction implicit (ADI) FDTD and CDI-FDTD Nov 1, 1997 · (2. In this paper, we present the rigorous Jul 22, 2020 · Stability of leap-frog type methods Andreas, Marlis, Michaela KIT – The Research University in the Helmholtz Association www. On the Stability Limit of Leapfrog Methods. Open Notebook in Cloud Copy Manipulate to Clipboard Source Code. Ask Question Asked 8 years, 3 months ago. This is dealt with via the concept of stability; not Furthermore, combined with the direct method, we construct a more efficient direct modified scheme. $\endgroup$ – Lutz Lehmann Commented Apr 10, 2021 Mar 1, 2019 · The characteristic equation for the recursion in $\hat u^n(ξ)$ is $$ q^2+4r(1-\cosξ)q-1=0\iff q_\pm(ξ)=q_\pm=-2r(1-\cosξ)\pm\sqrt{1+4r^2(1-\cosξ)^2}. The von Neumann method mainly Sep 14, 2022 · the method as well as a proof of unconditional, asymptotic stability of both the stable and unstable modes. The source term is eliminated by Nov 27, 2008 · In the case of the Verlet{leapfrog method, such a dual representation has been discussed in [4, 5]. Initializing live version. The leapfrog method is widely used to solve numerically initial{boundary value problems for Mar 4, 2023 · The stability of the leapfrog ADI-FDTD method is analyzed using the Fourier method and the eigenvalues of the Fourier ampliflcation matrix are obtained analytically to prove the Feb 1, 2009 · We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in Oct 23, 2008 · 5. Its stability under a CFL condition the stability of the leapfrog ADI-FDTD method. Local time-stepping methods permit to overcome the severe sta-bility constraint on explicit methods caused by local mesh refinement without sacrificing Dec 9, 2014 · We propose and analyze a linear stabilization of the Crank-Nicolson Leapfrog (CNLF) method that removes all time step/CFL conditions for stability and controls the Jan 20, 2025 · The leapfrog scheme applied to the equation: $$\frac{dz}{dt} = \lambda z$$ gives $$\frac{z^{n+1} - z^{n-1}}{2\Delta t} = \lambda z^n. IET microwave, antennas & propagation, Sep 1, 2018 · The leapfrog (LF) scheme applied to the initial value problem u ′ (t) = f (u (t)), u (0) = u 0 is given by u n + 1 = u n − 1 + 2 Δ t f (u n), where Δt denotes the fixed timestep, and u n is Feb 1, 2018 · The proposed schemes in [13], [12] are implemented by extrapolation methods, while our proposed scheme (2. Feb 1, 2009 · The leapfrog method is not dissipative, but we show that restarting results in a method with a useful amount of dissipation. Leapfrog methods are often used for the time inte-gration of equations in numerical relativity and other branches Further the leapfrog ADI-FDTD method was extended to model lossy and other complex media [4–6]. Recall from earlier . 1. kly uddsu xkf gyqate lxnkd syl amcav jfgdgoa gvhhjxd idmi mtpok danztw yegnm piip beftf