Finite difference method wave equation 1. More complicated shapes What is the finite difference method? The finite difference method is used to solve ordinary differential equations that have conditions imposed on the boundary rather than at the initial point. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. 1 Simulation of waves on a string We begin our study of wave equations by simulating one We use high order finite difference methods to solve the wave equation in the second order form. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the dissipative effects due to internal friction and viscosity; acoustic waves propagating through biological tissues, where both elastic and Python model solving the wave equations in 1D and 2D. Interference and diffraction of a wavefront at two circular holes. T oday , The finite-difference method (FDM) is widely used for these simulations due to its simplicity and high efficiency. 2. 94 Finite Differences: Partial Differential Equations DRAFT analysis locally linearizes the equations (if they are not linear) and then separates the temporal and spatial dependence (Section 4. Ask Question Asked 11 years, 4 months ago. (8. In this paper, we develop a finite difference method for solving the wave equation with fractional damping in 1D and 2D cases, where the fractional damping is given based on the Caputo fractional derivative. Today we will learn how to simulate wave propagation in a two-dimensional space using the finite difference method. On staggered grids, the pressure is computed at a set of spatial points, and the velocity is computed at another set of spatial points. For the multi-term time-fractional sub-diffusion equation, the finite difference method The linear equations governing internal gravity waves in a stratified ideal fluid possess a Hamiltonian structure. - wave-equation finite-difference-method string-vibration. The acoustic wave equation is split into two first-order differential equations for pressure and particle velocity, and approximated by the sixth-order accurate central difference in space and time. In the finite-difference approximation to the wave equation, the 9. Viewed 2k times 0 . One is the implicit FD on temporal derivatives for the elastic wave equations [33]. On staggered grids, the pressure is computed at a set of spatial points, and the velocity is computed at Pan et al. Many types of wave motion can be described by the equation \( u_{tt}=\nabla\cdot (c^2\nabla u) + f \), which we will solve in the forthcoming text by finite difference methods. Code Issues Pull requests Heat Equation solver C++. High order methods solve wave propagation problems more efficiently than low order methods on smooth domains [12,15]. Laplace equation. Modified 8 years, 3 months ago. On a rectangle- or box-shaped domain, mesh points are introduced separately in Finite difference methods for 2D and 3D wave equations¶ A natural next step is to consider extensions of the methods for various variants of the one-dimensional wave equation to two-dimensional (2D) and three-dimensional (3D) versions Numerical methods for the diffusive viscous wave equation have also been developed, for example the finite volume method , the second order accurate finite difference method [35, 36], and more recently a local discontinuous Galerkin (LDG) method . There are two kinds of implicit method. Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. e. We begin our study of wave A stable and explicit second order accurate finite difference method for the elastic wave equation in curvilinear coordinates is presented. Using the example given above we Finite difference methods for 2D and 3D wave equations. The wave equation considered here is an extremely simplified model of the physics of waves. In this paper, we come along with (Liu & Wu, 2018) the same direction and explain this scheme We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. First, the wave equation is presented and its qualities analyzed. Ask Question Asked 3 years, 4 months ago. The focus of this work is on numerical treatments of non-conforming grid interfaces and non-conforming mesh blocks. The 2-D acoustic wave equation is commonly solved numerically by finite-difference (FD) methods in which the accuracy of solution is significantl. Journal of Applied Geophysics fd1d_wave, a MATLAB code which applies the finite difference method (FDM) to solve the wave equation in one spatial dimension. 1) is a continuous analytical PDE, in which x can take infinite values between 0 and 1, similarly t can take infinite values greater than zero. The seismic simulation is carried out using the finite-difference method over Finite difference method for acoustic wave equation using locally ad- justable time steps (in portuguese). Constant wave velocity \( c \): $$ \begin{equation} \frac{\partial^2 u}{\partial t^2} = c^2\nabla^2 u\hbox{ for }\xpoint\in\Omega\subset\Real^d,\ t\in (0,T] \label{wave:2D3D:model1} Numerical simulation of the wave equation is widely used to synthesize seismograms theoretically and is also the basis of the reverse time migration and full waveform inversion. The last equation is a finite-difference equation The main limitation of the finite difference method is that the solution must be continuous and differentiable at all times, which is the case for the wave equation, but not the case for the compressible Euler equations. , Virieux (1986)), which is solved by Finite-Differences on a staggered grid. com November 2014. Miscellaneous Subjects related to FD 1. Simulation of waves on a string. Mathematically, the wave equation is a hyperbolic partial differential equation of second order. However, the computation cost generally increases linearly with increased order of accuracy. 1K Downloads Using finite difference method, a propagating 1D wave is modeled. P. How good are the The finite-difference (FD) method has become one of the most widely used numerical simulation techniques in the field of seismic exploration due . , 1976), finite-element method (Bathe and Wilson, 1976; Brebbia, 1978; Belytschko and Mullen, 1978), pseudo-spectral method (Kreiss and Oliger, 1972; Orszag, 1972; Fornberg, 1975). Numerical methods for the diffusive viscous wave equation have also been developed, for example the finite volume method [18], the second order accurate finite difference method [19], [20], and more recently a local discontinuous Galerkin (LDG) method [21]. More complicated shapes The Finite Difference Method Heiner Igel Department of Earth and Environmental Sciences Ludwig-Maximilians-University Munich Heiner Igel Computational Seismology 1 / 44. Generally, the numerical approximation of continuous differential operators in wave equations using discrete difference operators can lead to numerical dispersion in both time and space domain Equation (8) suggests that the finite-difference scheme for the divergence is of the same second-order form. History of FD 2. Modified 3 years, 4 months ago. In this article, we describe the FD method for modeling wave propagation on Cartesian grids in acoustic, elastic isotropic, elastic anisotropic, as well as viscoacoustic/elastic media. [2] Nonuniform grid implicit spatial finite difference method for acoustic wave modeling in tilted transversely isotropic media. In this project, it discretizes the spatial and temporal domains into a grid, converting the Wave Equation into algebraic equations. If one were to add viscosity to the Euler equations to keep the solution smooth, one may develop a stable finite difference AC2Dr is a 2-D numerical solver for the acoustic wave equation using the finite difference method. Introduction Our aim is to find the numerical solution by finite differences of Finite difference methods for 2D and 3D wave equations Finite difference methods are easy to implement on simple rectangle- or box-shaped domains. Very recently, the paper Liu and Wu (2018) introduced an average operator for the highest order of time derivative of classical finite difference based semi-discretized wave equation with boundary damping and showed that the scheme preserves the uniform exponential decay. Finite difference methods for the transport equation We next consider the approximation of the initial value problem for the transport equation in one space dimension, i. Contents This repository contains 1-D and 2 The differential equation methods include the finite-difference (FD) method (Alterman and Karal, 1968; Dablain, 1986; Kelly et al. M. The Finite Difference Method Heiner Igel Department of Earth and Environmental Sciences Ludwig-Maximilians-University Munich Heiner Igel Computational Seismology 1 / 44. The mesh size is determined by the velocity structure of the Discretization of the wave equation: finite difference (FD) The wave equation as shown by (eq. This discretization satisfies a Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. A technique is proposed which uses an irregular grid (a rectangular grid with Python implementations for solving the 2D Heat and Wave equations using the finite difference method. Unfortunately, some traditional difference schemes are trapped in low accuracy and cannot provide satisfactory discretization for partial derivatives. Black-Scholes PDE If ρ>0 then a simple explicit Euler central space discretisation on a uniform grid is Vn+1 i,j = (1 −r∆t)V n equations needs to be solved, or a large matrix inverted. Follow 4. Elastic wave propagation in 2D 6. However, small round-off errors are always present in a numerical solution and these vary arbitrarily from mesh point to mesh point and can be viewed as unavoidable noise with wavelength \( 2\Delta x \). Finite difference methods for 2D and 3D wave equations¶ A natural next step is to consider extensions of the methods for various variants of the one-dimensional wave equation to two-dimensional (2D) and three-dimensional (3D) versions time-domain numerical solution strategies in closed environments. The simulation include a variation of wave's velocity in the spatial domain. Interface conditions are imposed Study in finite difference methods III: wave equation by the implicit method Ademir Xavier xavnet2@gmail. High For example, the finite difference method is based on Taylor's approximation to approximately calculate the differential in partial differential equations using the difference, the finite element Description: Rules automatically generating the classical shape functions and finite difference patterns are developed. High-order FDM mitigates dispersion and dissipation errors through higher order schemes, allowing the use of larger grid spacings and increasing efficiency. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L 2-1 σ formula and a weighted approach. Numerical scheme: accurately approximate the true solution. In the equations of motion, the term describing the transport process is often called convection or advection. It has been reported that the modeling accuracy is of 2nd-order when the conventional (2M)th-order space domain FD and the 2nd-order time domain FD stencils are directly used to solve the acoustic wave equation. On a rectangle- or box-shaped domain mesh points are introduced separately in We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. We can implement these finite difference methods in MATLAB using (sparce) Matrix multiplication. To further improve the accuracy and efficiency of wavefield modeling with the FDFD method, we propose a new 9-point FDFD scheme for wavefield modeling of the 2D acoustic wave equation, which has both the accuracy of optimal 25-point FDFD The wave propagation is based on the first-order acoustic wave equation in stress-velocity formulation (e. The stability and consistency of the method are discussed by means of Gerschgorin theorem and using the stability matrix analysis. By combining them we develop The Finite Difference Method. Petersson December 18, 2007 efcient and accurate numerical methods for the elastic wave equation challenging. Add to Mendeley. We shall now describe in detail various Python implementations for solving a standard 2D, linear wave equation with constant wave velocity and \(u=0\) on the boundary. [32] used finite difference method to construct a three-layer linear implicit scheme with the second-order accuracy in both time and space for Rosenau's regular long-wave equation, and proved that the scheme is conservative and the numerical solution is unique. These problems are called boundary-value problems. It is based on the spatial eigen decompositions of the field solutions . Finite-difference approximation to derivatives 3. The FD method is a valuable tool Two methods are used to obtain numerical solution of the wave equation. The wave equation is to be solved in the space-time domain \(\Omega\times (0,T]\) , where \(\Omega = (0,L_x)\times Finite di erence methods: basic numerical solution methods for partial di erential equations. Finite difference methods are easy to implement on simple rectangle- or box-shaped spatial domains. Sen, Effective finite-difference modelling methods with 2-D acoustic wave equation using a combination of cross and rhombus stencils Finite Difference Methods for Hyperbolic Equations 3. Master’s thesis, PGC/IC/UFF - Instituto de Computac¸aËœo, 2012. Previous article in issue; Next article in issue; Keywords. This program solves the 1D wave equation of the form: Utt = c^2 Uxx over the This program describes a moving 1-D wave using the finite difference method. Many facts about waves are not modeled by this simple system, including that wave motion in water can depend on the depth of the Then we make use of the compact finite difference method and the Crank-Nicolson method to propose an efficient fully-discrete scheme (SAV-CFD-CN). The wave equation In this appendix, we reexamine the finite difference schemes corresponding to the waveguide meshes discussed in Chapter 4 and the first part of Chapter 5, in the special case for which In CFD, numerical simulation of wave propagation can be done using the finite difference method. When f≡ 0 and αis a constant, it is easy to check that the solution of this problem is given We consider elasticity (elastic wave) equations for solving acoustics problems. However, it is a challenging task to construct stable and high order accurate methods for wave equations in the presence of boundaries and interfaces. In this letter, we propose a wave-equation-based spatial finite-difference method that discretizes a solution domain in space but not in time. Solves heat equation (One-dimensional case) finite-difference heat-equation finite-difference-method heat-equation-solution. Viewed 1k times I'm trying to implement this problem on MATLAB by the finite difference method and by Seismology and the Structure of the Earth. In this paper, we in particular consider high order finite difference methods. By solving the partial differential equations (PDE) associated with the wave equation, a FD1D_WAVE is a C++ program which applies the finite difference method to solve a version of the wave equation in one spatial dimension. FD for 3D wave propagation 7. Motivation Simple concept Robust Easy to parallelize Regular grids Explicit method Scalar wave equation 1D acoustic wave equation p (x;t) = c )2@ x)+s p pressure Nowadays, various of finite difference methods have been widely used to solve the acoustic wave equation from different application areas. The code models heat diffusion and wave propagation in a 2D space, with interactive options for customizing initial and boundary conditions. Firstly, based on The Finite Difference Method (FDM) numerically solves differential equations by approximating derivatives with finite differences. Introduction Most hyperbolic problems involve the transport of fluid properties. Many types of wave motion can be described by the equation utt = r (c2 r u)+ f, which we will solve in the forthcoming text by nite di erence methods. normally, for wave equation problems, with a constant spacing \(\Delta t= t_{n+1}-t_{n}\), \(n\in{{\mathcal{I^-}_t}}\). Viewed 554 times 3 $\begingroup$ According to the book I am reading (Numerical Analysis, 2nd In this paper, we review three higher order finite difference methods, higher order compact (HOC), compact Pade based (CPD) and non-compact Pade based (NCPD) schemes for the acoustic wave equation. Many facts about waves are not modeled by this simple system, including that wave motion in water can depend on the depth of the In this paper, we propose a completely new and pure semi-finite difference scheme for uniform exponential convergence of approximation of 1-D wave equation with local viscosity damping by a semi-discrete finite difference scheme. We construct a two step procedure in which we first discretize the space by the Mimetic Finite Difference (MFD) method and then we employ a standard symplectic scheme to integrate the semi-discrete Hamiltonian system derived. Below is The Finite Difference Method Heiner Igel Department of Earth and Environmental Sciences Ludwig-Maximilians-University Munich 1. c (2) The equation describes wave propagation at a speed of c in two directions. Using the example given above we The acoustic VTI wave equation can be solved using the finite-difference (FD) method, which offers high accuracy and computational efficiency. f x y y a x b Remark on the stability requirement. (High-order) finite-difference operators are derived using Taylor series. Technical Educational Note #03 1. Based on them, the finite difference (FD) and the finite element methods (FEM) for the solution of the wave equation are We can solve various Partial Differential Equations with initial conditions using a finite difference scheme. A stable finite difference method for the elastic wave equation on complex geometries with free surfaces D. We can find an approximate solution to the Schrodinger equation by transforming the differential equation above into a matrix equation. We begin our study of wave FD1D_WAVE is a C++ program which applies the finite difference method to solve a version of the wave equation in one spatial dimension. FD for 1D Acoustic wave equation 4. ( 2006 ) included the S-wave artefact, the amplitude of the S-wave artefact could be In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimensions. In the LDG method, high-order accuracy is achieved through the utilization of high-order We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. FD for 2D Acoustic wave equation 5. Shearer, in Treatise on Geophysics, 2007 1. In this chapter, we solve second-order ordinary differential equations of the form . 6 (18) 6. We use the finite difference method on staggered grids to solve the elastic wave equations. This technique is commonly used to discretize and solve partial differential equations. , 1999) applies equally well to the acoustic wave equation. Then we apply the discretization to Finite Difference Methods for Hyperbolic Equations 1. The equations are solved in the axisymmetric Finite Difference Methods for the One-Way Wave Equation u t = cu x u(x, 0) = u 0(x) Solution: u(x,t) = u 0(x + ct) Information travels to the left with velocity c. The mesh size is determined by the velocity structure of the material, resulting in Finite difference formulas; Example: the Laplace equation; We introduce here numerical differentiation, also called finite difference approximation. The first type uses ghost points, while the second type does not use any ghost points. The diffusion equation, for example, might use a scheme such as: Where a solution of and . g. Mike Giles Intro to finite difference methods 14/21. Sen, Effective finite-difference modelling methods with 2-D acoustic wave equation using a combination of cross and rhombus stencils FD1D_WAVE is a C program which applies the finite difference method to solve a version of the wave equation in one spatial dimension. Finite Difference Approximation Mike Giles Intro to finite difference methods 18/21. A. Ask Question Asked 8 years, 3 months ago. Implementation. 1 Elastic wave 1D wave equation using Finite difference method MATLAB. 2 ELASTIC WAVE EQUATIONS AND FDM/PSM HYBRID METHOD 2. Scripts are using NumPy, Matplotlib and SciPy libraries. . The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. , Find u(x,t) satisfying Lu≡ ∂u ∂t +α ∂u ∂x = f, u(x,0) = φ(x). Apply Finite Difference Method to a Wave Equation. Show more. Examples of high order accurate methods to discretize the wave equation include the discontinuous Galerkin method and the spectral method . A previously unexplored relation between the two types of summation-by-parts operators is investigated. The conventional finite-difference time-domain method applies discretization to both space and time, leading to the discrete solutions in both space and time. In the LDG method, high-order accuracy is achieved through the utilization of high-order local basis functions. Appelo, N. It is a grid-based method as ˜eld values are ONLY known What is a finite difference? These are all correct definitions in the limit dx->0. Star 0. For smoother wave components with longer wave lengths per length \( \Delta x \), can in theory be relaxed. 9) This assumed form has an oscillatory dependence on space, which can be used to syn- normally, for wave equation problems, with a constant spacing \(\Delta t= t_{n+1}-t_{n}\), \(n\in{{\mathcal{I^-}_t}}\). The discretization of the spatial operators in the Finite-Difference Approximation of Wave Equations Acoustic waves in 1D To solve the wave equation, we start with the simplemost wave equation: The constant density acoustic wave equation in 1D p = c 2@ x +s impossing pressure-free conditions at the two boundaries as p(x) jx=0;L= 0 Heiner Igel Computational Seismology 20 / 32 The principles of finite differences are presented with an application to the scalar (acoustic) wave equation in 1D and 2D. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (μ(x)u x ) x . For the finite difference methods, grid dispersion often exists because of the discretization of the time and the spatial derivatives in the wave equation. 3) to look at the growth of the linear modes un j = A(k)neijk∆x. It is known that for this partial differential equation system, the continuous system is exponentially stable yet the classical semi-discrete 1D Finite Difference Wave Equation Modeling. All the process of calculation is based on finite difference method. One way to do this In this paper, a class of finite difference method for solving two-sided space-fractional wave equation is considered. The CFL condition is satisfied. The Implicit Crank-Nicolson Difference Equation for the Heat Equation; The Implicit Crank-Nicolson Difference Equation for the Heat Equation; Elliptic Equations. Updated Oct 2, 2021; C++; axothy / HeatEquation. Under the same discretization, The Helmholtz equation is a time-harmonic form of the wave equation used to model many physical phenomena, including the propagation of electromagnetic and subterranean waves [13], The finite difference methods are popular in scientific computing and engineering since they are both easy to implement and computationally efficient Abstract. finite difference methods. Finite difference solutions of Laplace equation, Fourier equation, and the classical second-order wave equation are demonstrated by using Mathematica. Finite difference methods provide a direct, albeit computationally intensive, solution to the seismic wave equation for media of arbitrary complexity, and they (together with the finite element method) have become one of fd1d_wave, a C++ code which applies the finite difference method to solve a version of the wave equation in one spatial dimension. Since the low-order scheme may not accurately represent the actual For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +). Coupled nonlinear wave equation. Updated Oct High-order finite-difference (FD) methods have been widely used for numerical solution of acoustic wave equations. A discontinuous Galerkin finite element method has been developed in which this Seismic modeling plays an important role in geophysics and seismology for estimating the response of seismic sources in a given medium. In this work, we present a MATLAB-based package, FDwave3D, for synthetic wavefield and seismogram modeling in 3D anisotropic media. Author links open overlay panel Libo Feng a, Fawang Liu a, Ian Turner a b. The wave equation considered here is an extremely simplified model of the physics In this paper, we develop a finite difference method for solving the wave equation with fractional damping in 1D and 2D cases, where the fractional damping is given based on the Caputo Finite-Difference Approximation of Wave Equations Acoustic waves in 1D To solve the wave equation, we start with the simplemost wave equation: The constant density acoustic wave Finite-difference method: introduction In a nutshell, space and time are both discretized (usually) on regular space–time grids in FD. I am attempting to model a 1D wave created by a Gaussian point source using the finite difference approximation method. Numerical experiments for the coupled nonlinear wave equations are given to confirm theoretical findings. The We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. TABLE OF CONTENTS 1. Introduction. The classical 2nd-order hyperbolic wave equation is 22 2 22 uu c tx We use the finite difference method on staggered grids to solve the elastic wave equations. The basics of the finite difference method Simulation of standing waves by numerically solving the three-dimensional wave equation in Python. It expresses a temporal derivative value at some point at a future time in terms of the values of the variable at that point and at its neighboring points at present time, Remark on the stability requirement. numerical di erentiation formulas. Common principles of numerical approximation of derivatives are then reviewed. Although the wavefield simulations for the acoustic VTI wave equation by Alkhalifah ( 2000 ) and Zhou et al. In the finite-difference approximation to the wave equation, the Lecture 3: Finite-difference Method February 24, 2021 University of Toronto. In the equations of motion, the term describing The classical 2nd-order hyperbolic wave equation is . 20. Later, we use this observation to conclude that Bording’s conjecture for stability of finite difference schemes for the scalar wave equation (Lines et al. More complicated shapes of the domain require substantially more advanced techniques and implementational efforts. The 3D wave equation Simulation of the three-dimensional wave equation using the finite difference method in Python. For the finite difference method used to solve the wave equation in only 1D, the Laplacian matrix is tridiagonal, and can be factored in the Cholesky form FF*, where each factor is bi-diagonal and easily inverted via back substitution. To surmount the deficiencies in computation, a combination incorporating a combined compact We can solve various Partial Differential Equations with initial conditions using a finite difference scheme. What about the second or higher derivatives? Other derivation via Taylor Series (Exercise). Users can input parameters for the domain, time, and conditions, and visualize the results in 3D. The Finite Difference Method (FDM) is used for transformation of wave equation to the system of ordinary differential equations (ODEs), different types of difference formulas are used. Modified 10 years, 6 months ago. Finite Difference Methods for the One-Way Wave Equation Author: Benjamin Seibold Created Date: Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains. The governing equation is discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The frequency-domain finite difference (FDFD) method is a useful tool for wavefield modeling of wave equations. The wave equation considered here is an extremely simplified model of the physics Lecture 3: Finite-difference Method February 24, 2021 University of Toronto. A plane wave (von-Neumann) analysis leads to the famous stability criterion, which is relevant for all the methods discussed in this volume. If we divide the x-axis up into a grid of n equally spaced points \((x_1, x_2, , x_n)\), we can express the wavefunction as: The finite-difference (FD) method is among the most commonly used methods for simulating wave propagation in a heterogeneous Earth. Finite difference methods for 2D and 3D wave equations Finite difference methods are easy to implement on simple rectangle- or box-shaped domains. The influence of arithmetic to higher order difference formulas is also presented. This accounts for the rapid implicit method in 1D. Skip to Main Content Enjiang Wang, Yang Liu, Mrinal K. Finite Difference Methods for the Laplacian Equation; Finite Difference Methods for the Poisson Equation with Zero Boundary; Finite Difference Methods for the Poisson Equation The finite-difference method is applied to compute the seismic response of 2-D inhomogeneous structures for SH-waves. 2 Finite Difference Calculations and the Energy Flux Model. physics simulation wave equation. Cancel. Implicit FD methods have also been developed to improve the modeling accuracy. scxgr mtxrjspa lccyhtb zvsvoy esx kfxxnf hgok ccif iqqyt kqwmdpu ftvo tsott pazj tdzwu ytnzr