Numerical Differentiation and Finite Difference Operators

(Numerical Integration). Determine the coefficients \(A_0\), \(A_1\), and \(A_2\) that make the formula \[ \int _0^2 f(x)\,dx \approx A_0 f(0) + A_1 f(1) + A_2 f(2) \] exact for all polynomials of degree 3.

(Unstable Multistep Method). Consider the numerical method \[ y_{k+1} = 3y_k - 2y_{k-1} + \frac {h}{2}\left (f(x_k,y_k)-3f(x_{k-1},y_{k-1})\right ), \qquad k \ge 1. \] Illustrate with an example of a simple initial value problem that the above scheme is unstable.

Consider the boundary value problem \[ u''(x) = f(x),\quad \text {for }0 < x < 1, \] \[ u(0) = 0,\quad u(1) = 0, \]

where \(f\) is a smooth function on \([0, 1]\), and the finite difference scheme \[ \frac {U_{j-1}-2U_j+U_{j+1}}{h^2} = f(x_j), \qquad j = 1,2,\ldots ,m, \] for finding approximations \(U_j\) to \(u(x_j)\). Here, \(x_j = jh\) and \((m+1)h = 1\).

Analyze the accuracy, stability, and convergence properties of the scheme.

Numerical Integration

Determine a formula \[ \int _{-\pi }^{\pi } f(x)\cos (x)\,dx \approx A_0 f\left (-\frac {3}{4}\pi \right ) +A_1 f\left (-\frac {1}{4}\pi \right ) +A_2 f\left (\frac {1}{4}\pi \right ) +A_3 f\left (\frac {3}{4}\pi \right ) \] that is exact when \(f\) is a polynomial of degree 3.

Unstable Multistep Method

Consider the numerical method \[ y_{k+1}=3y_k-2y_{k-1} +\frac {h}{2}\left (f(x_k,y_k)-3f(x_{k-1},y_{k-1})\right ), \qquad k\ge 1 \] Illustrate with an example of a simple initial value problem that the above scheme is unstable.

Finite difference formulas

Given \(h>0\), compute weights \(v_0\) and \(\{w_j\}_{j=-1}^{1}\) for the following finite difference formula for the third derivative \(f^{(3)}(x)\): \[ f^{(3)}(x)\approx v_0f'(x)+\sum _{j=-1}^{1}w_j f(x+jh) \] \[ =v_0f'(x)+w_{-1}f(x-h)+w_0f(x)+w_1f(x+h). \] What is the order of accuracy of your formula?

Higher Order Methods

Consider the following problem for \(v(x)\) on \([0,1]\): \[ v''(x)=f(x), \tag {2} \] with \(v(0)=v(1)=0\). Let \(N\cdot h=1\) and define \(x_j=j\cdot h\) for \(j=0,1,\ldots ,N\). The finite-difference scheme \[ \Delta _h U_j^h\equiv \frac {1}{h^2}\left (U_{j-1}^h-2U_j^h+U_{j+1}^h\right ) = f_j^h, \] for \(j=1,\ldots ,N-1\) with \(U_0^h=U_N^h=0\) gives values \(U_j^h\) that approximate \(v(x_j)\) with an error of \(O(h^2)\). Here the superscript \(h\) is used to indicate the grid size for the solution. Show how to use this method to find a numerical solution \(W_j\) whose values approximate \(v(x_j)\) with an error of \(O(h^4)\) and a numerical solution \(Y_j\) whose values approximate \(v(x_j)\) with an error of \(O(h^6)\).

Finite difference formulas

Given \(h>0\), compute weights \(w_j\) for the following 5-point finite difference formula for the third derivative \(f^{(3)}(x)\): \[ f^{(3)}(x)\approx \sum _{j=-2}^{2}w_j f(x+jh) \] \[ = w_{-2}f(x-2h)+w_{-1}f(x-h)+w_0f(x)+w_1f(x+h)+w_2f(x+2h). \] What is the order of accuracy of your formula?

Unstable Multistep Method

Consider the numerical method \[ y_{p+1}=3y_p-2y_{p-1} +\frac {h}{2}\left (f(x_p,y_p)-3f(x_{p-1},y_{p-1})\right ), \qquad p\ge 1, \]

for the solution of the initial value problem \[ y'=f(x,y),\qquad y(x_0)=y_0. \] Illustrate with an example of a simple initial value problem that the above numerical scheme is unstable.

Numerical Integration

Find a formula of the form \[ \int _0^{2\pi } f(x)\,dx = A_1 f(0)+A_2 f(\pi ) \] that is exact for any function having the form \[ f(x)=c+d\cos (x). \]

Linear Multistep Methods

Construct an example of:

a) a consistent but not stable (not zero-stable) linear multistep method

b) a stable (zero-stable) but not consistent linear multistep method

What kind of behavior do you expect from the numerical solution produced by the methods in a) and in b)?

Numerical Quadrature

Show that there is no set of nodes \(x_1,x_2,\ldots ,x_n\) and coefficients \(\alpha _1,\alpha _2,\ldots ,\alpha _n\) such that the quadrature rule \[ \sum _{j=1}^n \alpha _j f(x_j) \] exactly equals to the integral \(\int _a^b f(x)w(x)dx\) for all polynomials \(f(x)\) of degree less than or equal to \(2n\). \(w(x)\) is the weight function.

Local Truncation Error What is the local truncation error of a numerical scheme? What does consistency mean? Give a specific and precise example.

Numerical PDEs Consider the one-dimensional heat equation: Find \(u(x,t)\) such that \[ u_t=u_{xx},\qquad t\geq 0,\qquad x\in [0,1],\qquad u(x,0)=f(x),\qquad u(0,t)=u(1,t)=0. \] Describe how this problem might be solved by applying the Method of Lines and Euler’s Method. Give formulas that could be used to write a suitable computer code.

Wave Equation Consider the Lax-Friedrichs scheme below for the PDE \(v_t+cv_x=0\), where you may assume that \(c>0\): \[ u_j^{n+1}=\frac {1}{2}(u_{j-1}^n+u_{j+1}^n) -\frac {ck}{2h}(u_{j+1}^n-u_{j-1}^n), \] where \(h\) is the spacestep and \(k\) is the timestep. Note that the scheme can also be written \[ u_j^{n+1}=u_j^n-\frac {ck}{2h}(u_{j+1}^n-u_{j-1}^n) +\frac {1}{2}(u_{j-1}^n-2u_j^n+u_{j+1}^n). \]

a) Analyze the stability of this scheme in terms of \(\alpha =\frac {ck}{h}\).

b) Recall that a modified equation for a scheme is the PDE satisfied, to a certain order, by a smooth solution \(u(x,t)\) of the difference scheme. To zeroth order in \(k\) and \(h\), the modified equation for the Lax-Friedrichs scheme is \(u_t+cu_x=0\). Derive the modified equation through terms of first order in \(k\) and \(h\). You may assume that \(\alpha \) is constant. Does this modified equation predict the same stability condition as your analysis in part a)? Explain your answer. What does the modified equation suggest about the numerical solution that will be obtained for the step-function initial data \(u(x,0)=1\) for \(x\le 0\) and \(u(x,0)=0\) for \(x>0\)?

Numerical ODEs

a) The difference equation \[ y_{n+k}+a_1y_{n+k-1}+\cdots +a_ky_n=0 \] can, after the substitution \(\mathbf {y}_j=(y_j,y_{j+1},\ldots ,y_{j+k-1})^T\), be written in the form \(\mathbf {y}_{n+1}=A\mathbf {y}_n\). What does the matrix \(A\) look like? Describe the condition for \(y\) to remain bounded in terms of \(A\).

b) The differential equation \[ y^{(k)}+a_1y^{(k-1)}+\cdots +a_ky=0 \] can, after the substitution \(\mathbf {y}=(y,y',\ldots ,y^{(k-1)})^T\), be written in the form \(\mathbf {y}'=A\mathbf {y}\). What does the matrix \(A\) look like? Describe the condition for \(y\) to remain bounded in terms of \(A\).

c) Show that all solutions of the difference equation \[ y_{n+1}-2\lambda y_n+y_{n-1}=0 \] are bounded, as \(n\to \infty \), if \(\lambda \) is real and \(\lvert \lambda \rvert <1\).

Finite-Difference Methods Consider the initial boundary value problem \[ v_t=\beta v_{xx},\qquad 0<x<1,\quad t>0,\qquad v(x,0)=f(x) \] with Dirichlet boundary conditions \[ v(0,t)=0,\qquad v(1,t)=0. \] Suppose that the interval \([0,1]\) is partitioned by meshpoints \(x_j=jh\) for \(j=0,1,\ldots ,N\) and \(h=1/N\). A possible discretization of the above problem uses the Crank-Nicolson scheme for the differential equation: \[ \left (I-\frac {\beta k}{2}D_+D_-\right )u_j^{n+1} = \left (I+\frac {\beta k}{2}D_+D_-\right )u_j^n \] for \(j=1,\ldots ,N-1\), combined with the boundary conditions: \[ u_0^n=0\qquad \text {and}\qquad u_N^n=0 \] for each time level \(n\). Initial data is obtained by setting \(u_j^0=f_j\) for \(j=0,1,\ldots ,N\). (Note that \(D_+\) and \(D_-\) are the usual forward and backward difference operators.) Set up the matrix system that must be solved in order to advance the numerical solution from time level \(n\) to time level \(n+1\). Show that the suggested discretization is stable in the \(\ell _2\) norm \[ \lVert u\rVert _2=\left (\sum _{j=0}^N h\lvert u_j\rvert ^2\right )^{1/2} \] for all choices of timestep \(k\) and spacestep \(h\). (Note that von Neumann analysis is not enough to prove stability for the initial boundary value problem.) What other issues are important to consider in deciding whether this scheme is a good choice for the problem?

Fact 1: A real symmetric \(n\times n\) matrix \(A\) can be diagonalized by an orthogonal similarity transformation, and \(A\)’s eigenvalues are real.

Fact 2: The \((N-1)\times (N-1)\) matrix \(M\) defined by \[ M_{ij}= \begin {cases} -2, & i=j,\\ 1, & \lvert i-j\rvert =1,\\ 0, & \text {otherwise} \end {cases} \qquad (1\le i,j\le N-1) \] has eigenvalues \[ \mu _l=-4\sin ^2\left (\frac {\pi l}{2N}\right ),\qquad l=1,2,\ldots ,N-1. \]

Fact 3: The \((N+1)\times (N+1)\) matrix \(M\) defined by \[ M_{ij}= \begin {cases} -1, & i=j\in \{0,N\},\\ -2, & i=j,\ 1\le i\le N-1,\\ 1, & \lvert i-j\rvert =1,\\ 0, & \text {otherwise} \end {cases} \qquad (0\le i,j\le N) \] has eigenvalues \[ \mu _l=-4\sin ^2\left (\frac {\pi l}{2(N+1)}\right ),\qquad l=0,1,\ldots ,N. \]

Fact 4: For a real \(n\times n\) matrix \(A\), the Rayleigh quotient of a vector \(x\in \mathbb {R}^n\) is the scalar \[ r(x)=\frac {x^T Ax}{x^T x}. \] The gradient of \(r(x)\) is \[ \nabla r(x)=\frac {2}{x^T x}(Ax-r(x)x). \] If \(x\) is an eigenvector of \(A\) then \(r(x)\) is the corresponding eigenvalue and \(\nabla r(x)=0\).

Numerical ODEs

a) (8%) Consider the general system of ODEs for \(y(x)\in \mathbb {R}^n\), \(x\in \mathbb {R}\), and \[ f=[f_1(x,y),\ldots ,f_n(x,y)]^T: \qquad y'=f(x,y), \] where we assume \(f\) is continuously differentiable in both \(x\) and \(y\). To study local stability, we compare two solutions \(y_1(x)\) and \(y_2(x)\) where \(y_1(x_0)\approx y_2(x_0)\). Derive the linearized system for \(e(x)=y_1(x)-y_2(x)\) near \(x=x_0\) to show that \[ e'\approx Ae \] and find this square matrix \(A\). What is the most important property of \(A\) we need to find that will affect the choice of numerical schemes and step sizes?

b) (12%) Now consider the initial value problem \[ y''+2\mu y'+(1+\mu ^2)y=0, \] \[ y(0)=0,\qquad y'(0)=1, \] where \(\mu >0\) is a given constant. If we use a simple explicit Euler’s method, determine the maximum step size \(h\) that is allowed for the solution to be stable.

Iterative Methods for Linear Systems

a) (6%) Consider the linear system \(Ax=b\) and a splitting of \(A:A=M-K\), with \(M\) being nonsingular, an iterative method is to iterate \[ x_{m+1}=Rx_m+c, \] where \(R=M^{-1}K\) and \(c=M^{-1}b\). Show that the method converges for any \(x_0\) if \(\lVert R\rVert <1\), where \(\lVert \cdot \rVert \) is any operator norm.

b) (14%) To solve the boundary value problem \[ -u''(x)=f(x),\qquad \text {for }0<x<1, \] with \(u'(0)=0\) and \(u(1)=0\), we use second-order central differences for the discretization of the equation at mesh points \(x_j=j/N\): \[ -u_{j-1}+2u_j-u_{j+1}=f_j=\frac {f(x_j)}{N^2}, \qquad j=0,1,\ldots ,N-1 \] and the central difference approximation for \(u'\) at \(x_0=0\): \(u_1-u_{-1}=0\), to eliminate the ghost point.

Show that the finite difference equations and the boundary conditions lead to the following linear system \(Au=b\), where \[ \setlength {\arraycolsep }{5pt} A= \begin {bmatrix} 1 & -1 & 0 & \cdots & 0 & 0\\ -1 & 2 & -1 & \cdots & 0 & 0\\ 0 & -1 & 2 & \ddots & 0 & 0\\ \vdots & \vdots & \ddots & \ddots & -1 & 0\\ 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & -1 & 2 \end {bmatrix}, \quad u= \begin {pmatrix} u_0\\ u_1\\ \vdots \\ u_{N-2}\\ u_{N-1} \end {pmatrix}, \quad b=\frac {1}{N^2} \begin {pmatrix} \frac {1}{2}f_0\\ f_1\\ \vdots \\ f_{N-2}\\ f_{N-1} \end {pmatrix}. \] Notice that equations are scaled so that \(A\) is symmetric. Determine if \(A\) has eigenvectors \[ v^{(k)}= \begin {bmatrix} 1\\ \cos \frac {1}{2}k\pi x_1\\ \vdots \\ \cos \frac {1}{2}k\pi x_j\\ \vdots \\ \cos \frac {1}{2}k\pi x_{N-2}\\ \cos \frac {1}{2}k\pi x_{N-1} \end {bmatrix}, \qquad k=1,3,5,\ldots ,2N-3,2N-1, \] and use them to find the eigenvalues of the iteration matrix \(R\) for the Jacobi method. Analyze the convergence properties of the Jacobi method applied to this problem, and express the convergence speed as a function of \(N\). How does the number of iterations required to reduce the initial error by a factor of \(\delta \) depend on \(N\)? In reality, should you use an iterative method to solve this matrix problem? If not, what other method is preferred?

Wave Equations Consider the standard wave equation in one space dimension: \[ u_{tt}=c^2u_{xx},\qquad 0<x<1,\quad t>0, \] with initial conditions \[ u(x,0)=f(x),\qquad u_t(x,0)=g(x),\qquad 0<x<1 \] for given smooth functions \(f\) and \(g\), and boundary condition \[ u(0,t)=u(1,t)=0,\qquad t>0. \] The leap-frog scheme takes advantage of the fact that the equation can be reformulated as a system of first order equations \[ u_t+cv_x=0, \qquad v_t+cu_x=0. \] Denote \(x_j=j\Delta x\), \(U_j^n\approx u(x_j,n\Delta t)\), and \(V_{j+1/2}^{n+1/2}\approx v(x_{j+1/2},(n+\frac {1}{2})\Delta t)\); use central difference approximations for both derivatives to derive the leap-frog scheme for the system of equations. How do you implement the initial and boundary conditions for both \(u\) and \(v\)? Derive the truncation error for this approximation and determine the order of truncation error. For stability, consider solutions in the form \[ \begin {bmatrix} U^n\\ V^{n+1/2} \end {bmatrix} = \lambda ^n e^{ikx} \begin {bmatrix} \widetilde {U}\\ \widetilde {V} \end {bmatrix}, \] for some constants \(\widetilde {U}\) and \(\widetilde {V}\). Substitute this solution into the finite difference equations, find out what condition \(\lambda \) has to satisfy in order for \(\widetilde {U}\) and \(\widetilde {V}\) to have nontrivial solutions. For those \(\lambda \) values, compute \(\lvert \lambda \rvert \) to see if the stability condition \(\lvert \lambda \rvert \le 1\) is satisfied.

Numerical ODEs

(a) Consider the general system of ODEs for \(y(x)\in \mathbb {R}^n\), \(x\in \mathbb {R}\) and \(f=[f_1,\ldots ,f_n]^T\): \[ y'=f(x,y), \] where we assume \(f\) to be Lipschitz continuous. To study the numerical stability, we linearize \(f\) near \(x=x_0,y=y_0\) to obtain: \[ y'\approx f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+\nabla _y f(x_0,y_0)(y-y_0). \] If we denote \(A=\nabla _y f(x_0,y_0)\), then we claim that the local stability of the above system can be analyzed by studying the following constant coefficient system \[ y'=Ay. \] Justify this claim by analyzing the simple scalar equation \[ y'=a+bx+cy, \] where \(a,b,c\) are constants, in the explicit Euler’s method, to show that the stability is determined by how we solve \(y'=cy\). What is the most important property of \(A\) we need to find that will affect the choice of numerical schemes and step sizes?

(b) Now consider the initial value problem \[ y''+2\mu y'+(1+\mu ^2)y=0, \] \[ y(0)=0,\qquad y'(0)=1, \] where \(\mu >0\) is a given constant. If we use a simple explicit Euler’s method, determine the maximum step size \(h\) that is allowed for the solution to be stable.

Iterative Methods for Linear Systems

(a) Consider the linear system \(Ax=b\) and a splitting of \(A:A=M-K\), with \(M\) being nonsingular, an iterative method is to iterate \[ x_{m+1}=Rx_m+c, \] where \(R=M^{-1}K\) and \(c=M^{-1}b\). Show that the method converges for any \(x_0\) if \(\lVert R\rVert <1\) where \(\lVert \cdot \rVert \) is any operator norm.

(b) To solve the boundary value problem \[ -u''(x)=f(x),\qquad \text {for }0<x<1, \] with \(u(0)=0\), and \(u'(1)=0\), we use second-order central differences for the discretization at the mesh points \(x_j=j/N\): \[ -u_{j-1}+2u_j-u_{j+1}=f_j=\frac {f(x_j)}{N^2}, \qquad j=1,2,\ldots ,N \] and the central difference for \(u'\) at \(x_N\) to eliminate the ghost point \(x_{N+1}\): \[ u_{N+1}=u_{N-1}. \] Use all the finite difference equations and the boundary conditions to obtain the linear system \(Au=b\), where \[ \setlength {\arraycolsep }{5pt} A= \begin {bmatrix} 2 & -1 & 0 & \cdots & 0 & 0\\ -1 & 2 & -1 & \cdots & 0 & 0\\ 0 & -1 & 2 & \ddots & 0 & 0\\ \vdots & \vdots & \ddots & \ddots & -1 & 0\\ 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & -2 & 2 \end {bmatrix}, \quad u= \begin {pmatrix} u_1\\ u_2\\ \vdots \\ u_{N-1}\\ u_N \end {pmatrix}, \quad b=\frac {1}{N^2} \begin {pmatrix} f_1\\ f_2\\ \vdots \\ f_{N-1}\\ f_N \end {pmatrix}. \]

Verify that \(A\) has eigenvectors \[ v^{(k)}= \begin {bmatrix} \sin \frac {k\pi }{2}x_1\\ \sin \frac {k\pi }{2}x_2\\ \cdot \\ \cdot \\ \sin \frac {k\pi }{2}x_j\\ \cdot \\ \cdot \\ \sin \frac {k\pi }{2}x_{N-1}\\ \sin \frac {k\pi }{2}x_N \end {bmatrix}, \qquad k=1,3,5,\ldots ,2N-3,2N-1, \] and use them to find the eigenvalues of the iteration matrix \(R\) for the Jacobi method. Analyze the convergence properties of the Jacobi method applied to this problem, and express the convergence speed as a function of \(N\). How does the number of iterations required to reduce the initial error by a factor of \(\delta \) depend on \(N\)?

Wave Equations Consider the first-order wave equation in one space dimension: \[ u_t+au_x=0,\qquad -\infty <x<\infty ,\qquad t>0, \] where \(a\) is a constant, with the initial condition \[ u(x,0)=u_0(x),\qquad -\infty <x<\infty \] for a given function \(u_0\). We use the usual discretization \[ U_j^n\approx u(x_j,t^n), \] where \(x_j=j\Delta x\) and \(t^n=n\Delta t\). For a finite difference scheme to solve this problem, what is the CFL condition? Give an intuitive explanation for the condition. Derive the first-order upwind scheme for this equation and its truncation error. For numerical stability, derive the conditions for stability in the maximum norm (maximum principle) and the \(\ell _2\) norm (Fourier analysis).

Finite-Difference Methods Consider the initial boundary value problem \[ v_t=\beta v_{xx},\qquad 0<x<1,\qquad 0<t,\qquad v(x,0)=f(x) \] with Dirichlet boundary conditions \[ v(0,t)=0,\qquad v(1,t)=0. \]

Suppose that the interval \([0,1]\) is partitioned by meshpoints \(x_j=jh\) for \(j=0,1,\ldots ,N\) and \(h=1/N\). A possible discretization of the above problem uses the Crank-Nicolson scheme for the differential equation: \[ \left (I-\frac {\beta k}{2}D_+D_-\right )u_j^{n+1} = \left (I+\frac {\beta k}{2}D_+D_-\right )u_j^n \] for \(j=1,\ldots ,N-1\), combined with the boundary conditions: \[ u_0^n=0\qquad \text {and}\qquad u_N^n=0 \] for each time level \(n\). Initial data is obtained from by setting \(u_j^0=f_j\) for \(j=0,1,\ldots ,N\). (Note that \(D_+\) and \(D_-\) are the usual forward and backward difference operators.)

Set up the matrix system that must be solved in order to advance the numerical solution from time level \(n\) to time level \(n+1\). Show that the suggested discretization is stable in the \(l_2\) norm \[ \lVert u\rVert _2= \left (\sum _{j=0}^N h\lvert u_j\rvert ^2\right )^{1/2} \] for all choices of timestep \(k\) and spacestep \(h\). What other issues should be considered in deciding whether the suggested scheme is a good choice for the problem?

Local Truncation Error. What is the local truncation error of a numerical scheme? What does consistency mean? Give a specific and precise example.

Stability. What does stability of a numerical scheme mean? Give a specific example.

Finite-Difference Methods Consider the spatially-periodic initial value problem for \(v(x,t)\): \[ v_t=cv_x,\qquad \text {for }0\le x\le 1 \] with \[ v(x,0)=f(x),\qquad \text {for }0\le x\le 1. \] Here the function \(f\) has period 1, and \(c\) is a positive constant. Note that the exact solution of the problem is \(v(x,t)=f(x+ct)\).

Consider the following difference approximations to the PDE: \[ \begin {aligned} \text {i)}\quad &u_j^{n+1}=u_j^n+\frac {ck}{h}(u_{j+1}^n-u_j^n),\\ \text {ii)}\quad &u_j^{n+1}=u_j^n+\frac {ck}{h}(u_j^n-u_{j-1}^n),\\ \text {iii)}\quad &u_j^{n+1}=u_j^n+\frac {ck}{2h}(u_{j+1}^n-u_{j-1}^n). \end {aligned} \] In each case, \(Nh=1\), and the difference scheme is supplemented with the discrete initial conditions \(u_j^0=f(jh)\) for \(j=0,1,\ldots ,N\) and boundary conditions \(u_N^n=u_0^n\), \(u_{-1}^n=u_{N-1}^n\), and \(u_{N+1}^n=u_1^n\) for all time levels \(n\ge 0\). For each scheme, tell whether it is a reasonable scheme to use on the above problem, and justify your answers.

Compare the stability of the Euler Predictor-Trapezoidal Corrector and Backwards Euler timestepping methods for solution of the heat equation \[ u_t=u_{xx}\qquad u(x,0)=u_0(x) \] using the centered second difference operator \[ u_j \to \frac {u_{j+1}-2u_j+u_{j-1}}{h^2}\qquad j=1,\ldots ,J \] where \(u_j\approx u(jh)\) on the interval \([0,2\pi ]\) with periodic boundary conditions, and mesh-size \(h=2\pi /J\). Discuss the accuracy of these methods?