Exercises

Exercise 43: Geometric construction of the Forward Euler method

The section Understanding the Forward Euler method describes a geometric interpretation of the Forward Euler method. This exercise will demonstrate the geometric construction of the solution in detail. Consider the differential equation \( u'=u \) with \( u(0)=1 \). We use time steps \( \Delta t = 1 \).

a) Start at \( t=0 \) and draw a straight line with slope \( u'(0)=u(0)=1 \). Go one time step forward to \( t=\Delta t \) and mark the solution point on the line.

Solution.

b) Draw a straight line through the solution point \( (\Delta t, u^1) \) with slope \( u'(\Delta t)=u^1 \). Go one time step forward to \( t=2\Delta t \) and mark the solution point on the line.

Solution.

c) Draw a straight line through the solution point \( (2\Delta t, u^2) \) with slope \( u'(2\Delta t)=u^2 \). Go one time step forward to \( t=3\Delta t \) and mark the solution point on the line.

Solution.

d) Set up the Forward Euler scheme for the problem \( u'=u \). Calculate \( u^1 \), \( u^2 \), and \( u^3 \). Check that the numbers are the same as obtained in a)-c).

Solution.

% Forward Euler for u' = u
a = 0; b = 3;
dudt = @(u) u;
u_exact = @(t) exp(t);

u = zeros(4, 1);
u(1) = 1;
dt = 1;

for i = 1:3
    u(i+1) = u(i) + dt*dudt(u(i));
end

u
tP = [0 1 2 3];
time = linspace(a, b, 100);
u_true = u_exact(time);
plot(time, u_true, 'b-', tP, u, 'ro');
xlabel('t');
ylabel('u(t)');

Filename: ForwardEuler_geometric_solution.m.

Exercise 44: Make test functions for the Forward Euler method

The purpose of this exercise is to make a file test_ode_FE.m that makes use of the ode_FE function in the file ode_FE.m and automatically verifies the implementation of ode_FE.

a) The solution computed by hand in Exercise 43: Geometric construction of the Forward Euler method can be used as a reference solution. Make a function test_ode_FE_1() that calls ode_FE to compute three time steps in the problem \( u'=u \), \( u(0)=1 \), and compare the three values \( u^1 \), \( u^2 \), and \( u^3 \) with the values obtained in Exercise 43: Geometric construction of the Forward Euler method.

Solution.

b) The test in a) can be made more general using the fact that if \( f \) is linear in \( u \) and does not depend on \( t \), i.e., we have \( u'=ru \), for some constant \( r \), the Forward Euler method has a closed form solution as outlined in the section Derivation of the model: \( u^n=U_0(1+r\Delta t)^n \). Use this result to construct a test function test_ode_FE_2() that runs a number of steps in ode_FE and compares the computed solution with the listed formula for \( u^n \).

Solution.

function growth2_test()
    test_ode_FE_1();
    test_ode_FE_2();
end

function test_ode_FE_1()
    hand = [1 2 4 8];
    T = 3;
    dt = 1;
    U_0 = 1.0;
    function result = f(u, t)
        result = u;
    end
    [u, t] = ode_FE(@f, U_0, dt, T);
    tol = 1E-14;
    for i = 1:length(hand)
        err = abs(hand(i) - u(i));
        assert(err < tol, 'i=%d, err=%g', i, err);
    end
end
        
function test_ode_FE_2()
    T = 3;
    dt = 0.1;       % Tested first dt = 1.0, ok
    U_0 = 1.0; 
    function result = f(u, t)
        r = 1;
        result = r*u;
    end
    function result = u_exact(U_0, dt, n)
        r = 1;
        result = U_0*(1+r*dt)^n;
    end
    [u, t] = ode_FE(@f, U_0, dt, T);
    tol = 1E-12;     % Relaxed from 1E-14 to get through
    for n = 1:length(u)
        err = abs(u_exact(U_0, dt, n-1) - u(n));
        assert(err < tol, 'n=%d, err=%g', n, err);
    end
end

Filename: test_ode_FE.m.

Exercise 45: Implement and evaluate Heun's method

a) A 2nd-order Runge-Kutta method, also known has Heun's method, is derived in the section The 2nd-order Runge-Kutta method (or Heun's method). Make a function ode_Heun(f, U_0, dt, T) (as a counterpart to ode_FE(f, U_0, dt, T) in ode_FE.m) for solving a scalar ODE problem \( u'=f(u,t) \), \( u(0)=U_0 \), \( t\in (0,T] \), with this method using a time step size \( \Delta t \).

Solution.

function [u_found, t_used] = ode_Heun(f, U_0, dt, T)
    N_t = floor(T/dt);
    u = zeros(N_t+1, 1);
    t = linspace(0, N_t*dt, length(u));
    u(1) = U_0;
    for n = 1:N_t
        u_star = u(n) + dt*f(u(n),t(n));
        u(n+1) = u(n) + 0.5*dt*(f(u(n),t(n)) + f(u_star,t(n)));
    end
    u_found = u;
    t_used = t;
end

b) Solve the simple ODE problem \( u'=u \), \( u(0)=1 \), by the ode_Heun and the ode_FE function. Make a plot that compares Heun's method and the Forward Euler method with the exact solution \( u(t)=e^t \) for \( t\in [0,6] \). Use a time step \( \Delta t = 0.5 \).

Solution.

function demo_ode_Heun()
    % Test case: u' = u, u(0) = 1
    function value = f(u,t)
        value = u;
    end
    U_0 = 1;
    dt = 0.5;
    T = 6;
    [u_Heun, t] = ode_Heun(@f, U_0, dt, T);
    [u_FE, t]   = ode_FE(@f, U_0, dt, T);

    plot(t, u_Heun, 'b-', t, u_FE, 'b--', t, exp(t), 'r--');
    xlabel('t');
    legend('H', 'FE', 'true', 'location', 'northwest');
end

c) For the case in b), find through experimentation the largest value of \( \Delta t \) where the exact solution and the numerical solution by Heun's method cannot be distinguished visually. It is of interest to see how far off the curve the Forward Euler method is when Heun's method can be regarded as "exact" (for visual purposes).

Solution.

Filename: ode_Heun.m.

Exercise 46: Find an appropriate time step; logistic model

Compute the numerical solution of the logistic equation for a set of repeatedly halved time steps: \( \Delta t_k = 2^{-k}\Delta t \), \( k=0,1,\ldots \). Plot the solutions corresponding to the last two time steps \( \Delta t_{k} \) and \( \Delta t_{k-1} \) in the same plot. Continue doing this until you cannot visually distinguish the two curves in the plot. Then one has found a sufficiently small time step.

Hint.

Solution.

dt = 20; T = 100;
f = @(u,t) 0.1*(1-u/500)*u;
U_0 = 100;
[u_old, t_old] = ode_FE(f, U_0, dt, T);

k = 1;
one_more = true;
while one_more == true
    dt_k = 2^(-k)*dt;
    [u_new, t_new] = ode_FE(f, U_0, dt_k, T);
    plot(t_old, u_old, 'b-', t_new, u_new, 'r--')
    xlabel('t'); ylabel('N(t)'); 
    fprintf('Timestep was: %g\n', dt_k);
    answer = input('Do one more with finer dt (y/n)? ', 's')
    if strcmp(answer,'y')
        u_old = u_new;
        t_old = t_new;
    else
        one_more = false;
    end
    k = k + 1;
end

Filename: logistic_dt.m.

Exercise 47: Find an appropriate time step; SIR model

Repeat Exercise 46: Find an appropriate time step; logistic model for the SIR model.

Hint.

Solution.

function demo_SIR()
    % Find appropriate time step for an SIR model
    function result = f(u,t)
        beta = 10/(40*8*24);
        gamma = 3/(15*24);        
        S = u(1); I = u(2); R = u(3);
        result = [-beta*S*I beta*S*I - gamma*I gamma*I];
    end
    
    dt = 48.0;   % 48 h
    D = 30;      % Simulate for D days
    N_t = floor(D*24/dt);    % Corresponding no of hours
    T = dt*N_t;              % End time
    U_0 = [50 1 0];
    
    [u_old, t_old] = ode_FE(@f, U_0, dt, T);
    
    S_old = u_old(:,1);
    I_old = u_old(:,2);
    R_old = u_old(:,3);
    k = 1;
    one_more = true
    while one_more == true
        dt_k = 2^(-k)*dt;
        [u_new, t_new] = ode_FE(@f, U_0, dt_k, T);
        S_new = u_new(:,1);
        I_new = u_new(:,2);
        R_new = u_new(:,3);

        plot(t_old, S_old, 'b-', t_new, S_new, 'b--',...
             t_old, I_old, 'r-', t_new, I_new, 'r--',...
             t_old, R_old, 'g-', t_new, R_new, 'g--');
        xlabel('hours'); 
        ylabel('S (blue), I (red), R (green)');
        fprintf('Finest timestep was: %g\n', dt_k);
        answer = input('Do one more with finer dt (y/n)? ','s');
        if strcmp(answer, 'y')
            S_old = S_new;
            R_old = R_new;
            I_old = I_new;
            t_old = t_new;
            k = k + 1;
        else
            one_more = false;
        end
    end
end

Filename: SIR_dt.m.

Exercise 48: Model an adaptive vaccination campaign

In the SIRV model with time-dependent vaccination from the section Discontinuous coefficients: a vaccination campaign, we want to test the effect of an adaptive vaccination campaign where vaccination is offered as long as half of the population is not vaccinated. The campaign starts after \( \Delta \) days. That is, \( p=p_0 \) if \( V < \frac{1}{2}(S^0+I^0) \) and \( t>\Delta \) days, otherwise \( p=0 \).

Demonstrate the effect of this vaccination policy: choose \( \beta \), \( \gamma \), and \( \nu \) as in the section Discontinuous coefficients: a vaccination campaign, set \( p=0.001 \), \( \Delta =10 \) days, and simulate for 200 days.

Hint.

Solution.

function SIRV_p_adapt()
    % As SIRV1.py, but including time-dependent vaccination. 
    % Time unit: 1 h
    
    beta = 10/(40*8*24);
    beta = beta/4;        % Reduce beta compared to SIR1.py
    fprintf('beta: %g\n', beta);
    gamma = 3/(15*24);
    dt = 0.1;             % 6 min
    D = 200;              % Simulate for D days
    N_t = floor(D*24/dt); % Corresponding no of hours
    nu = 1/(24*50);       % Average loss of immunity
    Delta = 10;           % Start point of campaign (in days)
    p_0 = 0.001;

    t = linspace(0, N_t*dt, N_t+1);
    S = zeros(N_t+1, 1);
    I = zeros(N_t+1, 1);
    R = zeros(N_t+1, 1);
    V = zeros(N_t+1, 1);
   
    function result = p(t, n)
        if (V(n) < 0.5*(S(1)+I(1)) && t > Delta*24)
            result = p_0;
        else
            result = 0;
        end
    end

    % Initial condition
    S(1) = 50;
    I(1) = 1;
    R(1) = 0;
    V(1) = 0;

    epsilon = 1e-12;
    % Step equations forward in time
    for n = 1:N_t
        S(n+1) = S(n) - dt*beta*S(n)*I(n) + dt*nu*R(n) -...
                                          dt*p(t(n),n)*S(n);
        V(n+1) = V(n) + dt*p(t(n),n)*S(n);
        I(n+1) = I(n) + dt*beta*S(n)*I(n) - dt*gamma*I(n);
        R(n+1) = R(n) + dt*gamma*I(n) - dt*nu*R(n);
        loss = (V(n+1) + S(n+1) + R(n+1) + I(n+1)) -...
                       (V(1) + S(1) + R(1) + I(1));
        if loss > epsilon
            fprintf('loss: %g\n', loss);
        end
    end
    
    figure();
    plot(t, S, t, I, t, R, t, V);
    legend('S', 'I', 'R', 'V', 'Location', 'northeast');
    xlabel('hours');
    % saveas() gives poor resolution in plots, use the 
    % third party function export_fig to save the plot?
    
end

Filename: SIRV_p_adapt.m.

Exercise 49: Make a SIRV model with time-limited effect of vaccination

We consider the SIRV model from the section Incorporating vaccination, but now the effect of vaccination is time-limited. After a characteristic period of time, \( \pi \), the vaccination is no more effective and individuals are consequently moved from the V to the S category and can be infected again. Mathematically, this can be modeled as an average leakage \( -\pi^{-1}V \) from the V category to the S category (i.e., a gain \( \pi^{-1}V \) in the latter). Write up the complete model, implement it, and rerun the case from the section Incorporating vaccination with various choices of parameters to illustrate various effects. Filename: SIRV1_V2S.m.

Exercise 50: Refactor a flat program

Consider the file osc_FE.m implementing the Forward Euler method for the oscillating system model (4.43)-(4.44). The osc_FE.m is what we often refer to as a flat program, meaning that it is just one main program with no functions. To easily reuse the numerical computations in other contexts, place the part that produces the numerical solution (allocation of arrays, initializing the arrays at time zero, and the time loop) in a function osc_FE(X_0, omega, dt, T), which returns u, v, t. Place the particular computational example in osc_FE.m in a function demo(). Construct the file osc_FE_func.m such that the osc_FE function can easily be reused in other programs.

Solution.

function [u, v, t] = osc_FE(X_0, omega, dt, T)
    N_t = floor(T/dt);
    u = zeros(N_t+1, 1);
    v = zeros(N_t+1, 1);
    t = linspace(0, N_t*dt, N_t+1);

    % Initial condition
    u(1) = X_0;
    v(1) = 0;

    % Step equations forward in time
    for n = 1:N_t
        u(n+1) = u(n) + dt*v(n);
        v(n+1) = v(n) - dt*omega^2*u(n);
    end
end

function demo_osc_FE()
    omega = 2;
    P = 2*pi/omega;
    dt = P/20;
    T = 3*P;
    X_0 = 2;
    [u, v, t] = osc_FE(X_0, omega, dt, T);

    plot(t, u, 'b-', t, X_0*cos(omega*t), 'r--');
    legend('numerical', 'exact', 'Location', 'nortwest');
    xlabel('t');
end

Filename: osc_FE_func.m.

Exercise 51: Simulate oscillations by a general ODE solver

Solve the system (4.43)-(4.44) using the general solver ode_FE in the file ode_system_FE.m described in the section Programming the numerical method; the general case. Program the ODE system and the call to the ode_FE function in a separate file osc_ode_FE.m.

Equip this file with a test function that reads a file with correct \( u \) values and compares these with those computed by the ode_FE function. To find correct \( u \) values, modify the program osc_FE.m to dump the u array to file, run osc_FE.m, and let the test function read the reference results from that file.

Solution.

function test_osc_ode_FE()
    function result = f(sol, t)
        u = sol(1);  v = sol(2);
        %fprintf('dudt: %g , dvdt: %g\n', v, -omega^2*u);
        result = [v -omega^2*u];
    end
    
    % Set and compute problem dependent parameters
    omega = 2;   
    X_0 = 1;
    number_of_periods = 6;
    time_intervals_per_period = 20;

    P = 2*pi/omega;                     % Length of one period
    dt = P/time_intervals_per_period;   % Time step       
    T = number_of_periods*P;            % Final simulation time
    U_0 = [X_0 0];                      % Initial conditions
       
    % Produce u data on file that will be used for reference
    file_name = 'osc_FE_data';
    osc_FE_2file(file_name, X_0, omega, dt, T);    
    
    [sol, t] = ode_FE(@f, U_0, dt, T);
    u = sol(:,1);
       
    % Read data from file for comparison     
    infile = fopen(file_name, 'r');
    u_ref = fscanf(infile, '%f');
    fclose(infile);
        
    tol = 1E-5;               % Tried several stricter ones first
    for n = 1:length(u)
        err = abs(u_ref(n) - u(n));
        assert(err < tol, 'n=%d, err=%g', n, err);
    end
    
    % Choose to also plot, just to get a visual impression of u
    plot(t, u_ref, 'b-', t, u, 'r--');
    legend('u ref', 'u');
end

function osc_FE_2file(filename, X_0, omega, dt, T)
    N_t = floor(T/dt);
    u = zeros(N_t+1, 1);
    v = zeros(N_t+1, 1);

    % Initial condition
    u(1) = X_0;
    v(1) = 0;
    
    outfile = fopen(filename, 'w');
    fprintf(outfile,'%10.5f\n', u(1));

    % Step equations forward in time
    for n = 1:N_t
        u(n+1) = u(n) + dt*v(n);
        v(n+1) = v(n) - dt*omega^2*u(n);
        fprintf(outfile,'%10.5f\n', u(n+1));
    end
    fclose(outfile);  
end

Filename: osc_ode_FE.m.

Exercise 52: Compute the energy in oscillations

a) Make a function osc_energy(u, v, omega) for returning the potential and kinetic energy of an oscillating system described by (4.43)-(4.44). The potential energy is taken as \( \frac{1}{2}\omega^2u^2 \) while the kinetic energy is \( \frac{1}{2}v^2 \). (Note that these expressions are not exactly the physical potential and kinetic energy, since these would be \( \frac{1}{2}mv^2 \) and \( \frac{1}{2}ku^2 \) for a model \( mx'' + kx=0 \).)

Place the osc_energy in a separate file osc_energy.m such that the function can be called from other functions.

Solution.

function [U, K] = osc_energy(u, v, omega)
    U = 0.5*omega.^2*u.^2;
    K = 0.5*v.^2;
end

b) Add a call to osc_energy in the programs osc_FE.m and osc_EC.m and plot the sum of the kinetic and potential energy. How does the total energy develop for the Forward Euler and the Euler-Cromer schemes?

Solution.

omega = 2;
P = 2*pi/omega;
dt = P/20;
T = 3*P;
N_t = floor(round(T/dt));
t = linspace(0, N_t*dt, N_t+1);

u = zeros(N_t+1, 1);
v = zeros(N_t+1, 1);

% Initial condition
X_0 = 2;
u(1) = X_0;
v(1) = 0;

% Step equations forward in time
for n = 1:N_t
    u(n+1) = u(n) + dt*v(n);
    v(n+1) = v(n) - dt*omega^2*u(n);
end

[U, K] = osc_energy(u, v, omega);

plot(t, U+K, 'b-');
xlabel('t');
ylabel('U+K');

omega = 2;
P = 2*pi/omega;
dt = P/20;
T = 40*P;
N_t = floor(round(T/dt));
t = linspace(0, N_t*dt, N_t+1);

u = zeros(N_t+1, 1);
v = zeros(N_t+1, 1);

% Initial condition
X_0 = 2;
u(1) = X_0;
v(1) = 0;

% Step equations forward in time
for n = 1:N_t
    v(n+1) = v(n) - dt*omega^2*u(n);
    u(n+1) = u(n) + dt*v(n+1);
end

[U, K] = osc_energy(u, v, omega);

plot(t, U+K, 'b-');
xlabel('t');
ylabel('U+K');

Filenames: osc_energy.m, osc_FE_energy.m, osc_EC_energy.m.

Exercise 53: Use a Backward Euler scheme for population growth

We consider the ODE problem \( N'(t)=rN(t) \), \( N(0)=N_0 \). At some time, \( t_n=n\Delta t \), we can approximate the derivative \( N'(t_n) \) by a backward difference, see Figure 38: $$ N'(t_n)\approx \frac{N(t_n) - N(t_n-\Delta t)}{\Delta t} = \frac{N^n - N^{n-1}}{\Delta t},$$ which leads to $$ \frac{N^n - N^{n-1}}{\Delta t} = rN^n\thinspace,$$ called the Backward Euler scheme.

a) Find an expression for the \( N^n \) in terms of \( N^{n-1} \) and formulate an algorithm for computing \( N^n \), \( n=1,2,\ldots,N_t \).

Solution.

b) Implement the algorithm in a) in a function growth_BE(N_0, dt, T) for solving \( N'=rN \), \( N(0)=N_0 \), \( t\in (0,T] \), with time step \( \Delta t \) (dt).

Solution.

function N = growth_BE(r, N_0, dt, T)
    N_t = floor(round(T/dt));
    N = zeros(N_t+1, 1);
    N(1) = N_0;
    for n = 1:N_t+1
        N(n) = (1-r*dt)^(-n) * N_0;
    end
end

c) Implement the Forward Euler scheme in a function growth_FE(N_0, dt, T) as described in b).

Solution.

function N = growth_FE(r, N_0, dt, T)
    N_t = floor(round(T/dt));
    N = zeros(N_t+1, 1);
    N(1) = N_0;
    for n = 1:N_t+1
        N(n) = (1+r*dt)^n * N_0;
    end
end

function apply_growth_BE_and_FE()
    r = 1;
    N_0 = 1;   T = 6;
    dt = 0.5;
    N_BE = growth_BE(r, N_0, dt, T);
    N_FE = growth_FE(r, N_0, dt, T);
    time = linspace(0, T, floor(round(T/dt))+1);
    plot(time, N_BE, time, N_FE, time, exp(time));
    title('Timestep 0.5');
    legend('N (BE)', 'N (FE)', 'exp(t) (true)',...
                                 'Location', 'northwest');
    xlabel('t'); 
    print('tmp1', '-dpdf');  

    dt = 0.05;
    N_BE = growth_BE(r, N_0, dt, T);
    N_FE = growth_FE(r, N_0, dt, T);
    time = linspace(0, T, floor(round(T/dt))+1);
    figure(), plot(time, N_BE, time, N_FE, time, exp(time));
    title('Timestep 0.05');
    legend('N (BE)', 'N (FE)', 'exp(t) (true)',...
                                 'Location', 'northwest');
    xlabel('t'); 
    print('tmp2', '-dpdf');
end

d) Compare visually the solution produced by the Forward and Backward Euler schemes with the exact solution when \( r=1 \) and \( T=6 \). Make two plots, one with \( \Delta t = 0.5 \) and one with \( \Delta t=0.05 \).

Solution.

Filename: growth_BE.m.

Exercise 54: Use a Crank-Nicolson scheme for population growth

It is recommended to do Exercise 53: Use a Backward Euler scheme for population growth prior to the present one. Here we look at the same population growth model \( N'(t)=rN(t) \), \( N(0)=N_0 \). The time derivative \( N'(t) \) can be approximated by various types of finite differences. Exercise 53: Use a Backward Euler scheme for population growth considers a backward difference (Figure 38), while the section Numerical solution explained the forward difference (Figure 21). A centered difference is more accurate than a backward or forward difference: $$ N'(t_n+\frac{1}{2}\Delta t)\approx\frac{N(t_n+\Delta t)-N(t_n)}{\Delta t}= \frac{N^{n+1}-N^n}{\Delta t}\thinspace .$$ This type of difference, applied at the point \( t_{n+\frac{1}{2}}=t_n + \frac{1}{2}\Delta t \), is illustrated geometrically in Figure 39.

a) Insert the finite difference approximation in the ODE \( N'=rN \) and solve for the unknown \( N^{n+1} \), assuming \( N^n \) is already computed and hence known. The resulting computational scheme is often referred to as a Crank-Nicolson scheme.

Solution.

b) Implement the algorithm in a) in a function growth_CN(N_0, dt, T) for solving \( N'=rN \), \( N(0)=N_0 \), \( t\in (0,T] \), with time step \( \Delta t \) (dt).

Solution.

function N = growth_CN(r, N_0, dt, T)
    N_t = floor(round(T/dt));
    N = zeros(N_t+1, 1);
    N(1) = N_0;
    for n = 1:N_t
        N(n+1) = ( (1+0.5*dt*r)/(1-0.5*dt*r) ) * N(n);
    end
end

c) Make plots for comparing the Crank-Nicolson scheme with the Forward and Backward Euler schemes in the same test problem as in Exercise 53: Use a Backward Euler scheme for population growth.

Solution.

function apply_growth_CN()
    r = 1;
    N_0 = 1;   T = 6;
    dt = 0.5;
    N_CN = growth_CN(r, N_0, dt, T); 
    N_BE = growth_BE(r, N_0, dt, T);
    N_FE = growth_FE(r, N_0, dt, T);
    time = linspace(0, T, floor(round(T/dt))+1);
    plot(time, N_BE, time, N_FE, time, N_CN, time, exp(time));
    title('Timestep 0.5');
    legend('N (BE)', 'N (FE)', 'N (CN)', 'e^t (true)');
    xlabel('t'); 
    print('tmp1', '-dpdf');  

    dt = 0.05;
    N_CN = growth_CN(r, N_0, dt, T);
    N_BE = growth_BE(r, N_0, dt, T);
    N_FE = growth_FE(r, N_0, dt, T);
    time = linspace(0, T, floor(round(T/dt))+1);
    figure();
    plot(time, N_BE, time, N_FE, time, N_CN, time, exp(time));
    title('Timestep 0.05');
    legend('N (BE)', 'N (FE)', 'N (CN)', 'e^t (true)');
    xlabel('t'); 
    print('tmp2', '-dpdf');  
end

Filename: growth_CN.m.

Exercise 55: Understand finite differences via Taylor series

The Taylor series around a point \( x=a \) can for a function \( f(x) \) be written $$ \begin{align*} f(x) &= f(a) + \frac{d}{dx}f(a)(x-a) + \frac{1}{2!}\frac{d^2}{dx^2}f(a)(x-a)^2 \\ &\quad + \frac{1}{3!}\frac{d^3}{dx^3}f(a)(x-a)^3 + \ldots\\ & = \sum_{i=0}^\infty \frac{1}{i!}\frac{d^i}{dx^i}f(a)(x-a)^i\thinspace . \end{align*} $$ For a function of time, as addressed in our ODE problems, we would use \( u \) instead of \( f \), \( t \) instead of \( x \), and a time point \( t_n \) instead of \( a \): $$ \begin{align*} u(t) &= u(t_n) + \frac{d}{dt}u(t_n)(t-t_n) + \frac{1}{2!}\frac{d^2}{dt^2}u(t_n)(t-t_n)^2\\ &\quad + \frac{1}{3!}\frac{d^3}{dt^3}u(t_n)(t-t_n)^3 + \ldots\\ & = \sum_{i=0}^\infty \frac{1}{i!}\frac{d^i}{dt^i}u(t_n)(t-t_n)^i\thinspace . \end{align*} $$

a) A forward finite difference approximation to the derivative \( f'(a) \) reads $$ u'(t_n) \approx \frac{u(t_n+\Delta t)- u(t_n)}{\Delta t}\thinspace .$$ We can justify this formula mathematically through Taylor series. Write up the Taylor series for \( u(t_n+\Delta t) \) (around \( t=t_n \), as given above), and then solve the expression with respect to \( u'(t_n) \). Identify, on the right-hand side, the finite difference approximation and an infinite series. This series is then the error in the finite difference approximation. If \( \Delta t \) is assumed small (i.e. \( \Delta t < < 1 \)), \( \Delta t \) will be much larger than \( \Delta t^2 \), which will be much larger than \( \Delta t^3 \), and so on. The leading order term in the series for the error, i.e., the error with the least power of \( \Delta t \) is a good approximation of the error. Identify this term.

Solution.

b) Repeat a) for a backward difference: $$ u'(t_n) \approx \frac{u(t_n)- u(t_n-\Delta t)}{\Delta t}\thinspace .$$ This time, write up the Taylor series for \( u(t_n-\Delta t) \) around \( t_n \). Solve with respect to \( u'(t_n) \), and identify the leading order term in the error. How is the error compared to the forward difference?

Solution.

c) A centered difference approximation to the derivative, as explored in Exercise 54: Use a Crank-Nicolson scheme for population growth, can be written $$ u'(t_n+\frac{1}{2}\Delta t) \approx \frac{u(t_n+\Delta t)- u(t_n)}{\Delta t}\thinspace .$$ Write up the Taylor series for \( u(t_n) \) around \( t_n+\frac{1}{2}\Delta t \) and the Taylor series for \( u(t_n+\Delta t) \) around \( t_n+\frac{1}{2}\Delta t \). Subtract the two series, solve with respect to \( u'(t_n+\frac{1}{2}\Delta t) \), identify the finite difference approximation and the error terms on the right-hand side, and write up the leading order error term. How is this term compared to the ones for the forward and backward differences?

Solution.

d) Can you use the leading order error terms in a)-c) to explain the visual observations in the numerical experiment in Exercise 54: Use a Crank-Nicolson scheme for population growth?

Solution.

e) Find the leading order error term in the following standard finite difference approximation to the second-order derivative: $$ u''(t_n) \approx \frac{u(t_n+\Delta t)- 2u(t_n) + u(t_n-\Delta t)}{\Delta t}\thinspace .$$

Hint.

Solution.

Filename: Taylor_differences.pdf.

Exercise 56: Use a Backward Euler scheme for oscillations

Consider (4.43)-(4.44) modeling an oscillating engineering system. This \( 2\times 2 \) ODE system can be solved by the Backward Euler scheme, which is based on discretizing derivatives by collecting information backward in time. More specifically, \( u'(t) \) is approximated as $$ u'(t) \approx \frac{u(t) - u(t-\Delta t)}{\Delta t}\thinspace .$$ A general vector ODE \( u'=f(u,t) \), where \( u \) and \( f \) are vectors, can use this approximation as follows: $$ \frac{u^{n} - u^{n-1}}{\Delta t} = f(u^n,t_n),$$ which leads to an equation for the new value \( u^n \): $$ u^n -\Delta tf(u^n,t_n) = u^{n-1}\thinspace .$$ For a general \( f \), this is a system of nonlinear algebraic equations.

However, the ODE (4.43)-(4.44) is linear, so a Backward Euler scheme leads to a system of two algebraic equations for two unknowns: $$ \begin{align} u^{n} - \Delta t v^{n} &= u^{n-1}, \tag{4.86}\\ v^{n} + \Delta t \omega^2 u^{n} &= v^{n-1} \tag{4.87} \thinspace . \end{align} $$

a) Solve the system for \( u^n \) and \( v^n \).

Solution.

b) Implement the found formulas for \( u^n \) and \( v^n \) in a program for computing the entire numerical solution of (4.43)-(4.44).

Solution.

omega = 2;
P = 2*pi/omega;
dt = P/20;
%dt = P/2000;
T = 3*P;
N_t = floor(round(T/dt));
t = linspace(0, N_t*dt, N_t+1);

u = zeros(N_t+1, 1);
v = zeros(N_t+1, 1);

% Initial condition
X_0 = 2;
u(1) = X_0;
v(1) = 0;

% Step equations forward in time
for n = 2:N_t+1
    u(n) = (1.0/(1+(dt*omega)^2)) * (dt*v(n-1) + u(n-1));
    v(n) = (1.0/(1+(dt*omega)^2)) * (-dt*omega^2*u(n-1) + v(n-1));
end

plot(t, u, 'b-', t, X_0*cos(omega*t), 'r--');
legend('numerical', 'exact', 'Location', 'northwest');
xlabel('t');
print('tmp', '-dpdf');  print('tmp', '-dpng');

c) Run the program with a \( \Delta t \) corresponding to 20 time steps per period of the oscillations (see the section Programming the numerical method; the special case for how to find such a \( \Delta t \)). What do you observe? Increase to 2000 time steps per period. How much does this improve the solution?

Solution.

Filename: osc_BE.m.

Remarks

While the Forward Euler method applied to oscillation problems \( u''+\omega^2u=0 \) gives growing amplitudes, the Backward Euler method leads to significantly damped amplitudes.

Exercise 57: Use Heun's method for the SIR model

Make a program that computes the solution of the SIR model from the section Spreading of a flu both by the Forward Euler method and by Heun's method (or equivalently: the 2nd-order Runge-Kutta method) from the section The 2nd-order Runge-Kutta method (or Heun's method). Compare the two methods in the simulation case from the section Programming the numerical method; the special case. Make two comparison plots, one for a large and one for a small time step. Experiment to find what "large" and "small" should be: the large one gives significant differences, while the small one lead to very similar curves.

Solution.

% Time unit: 1 h
beta = 10.0/(40*8*24);
gamma = 3.0/(15*24);
%dt = 4.0;            % Small dt
dt = 24.0;            % Large dt
D = 30;               % Simulate for D days
N_t = floor(D*24/dt); % Corresponding no of hours

t = linspace(0, N_t*dt, N_t+1);
S_FE = zeros(N_t+1, 1); 
I_FE = zeros(N_t+1, 1); 
R_FE = zeros(N_t+1, 1);
S_H  = zeros(N_t+1, 1); 
I_H  = zeros(N_t+1, 1); 
R_H  = zeros(N_t+1, 1);

% Initial condition
S_FE(1) = 50; I_FE(1) = 1; R_FE(1) = 0;
S_H(1) = 50;  I_H(1) = 1;  R_H(1) = 0;

% Step equations forward in time
for n = 1:N_t
    % FE
    S_FE(n+1) = S_FE(n) - dt*beta*S_FE(n)*I_FE(n);
    I_FE(n+1) = I_FE(n) + dt*beta*S_FE(n)*I_FE(n) - dt*gamma*I_FE(n);
    R_FE(n+1) = R_FE(n) + dt*gamma*I_FE(n);
    % H  (predict and then update)
    S_p = S_H(n) - dt*beta*S_H(n)*I_H(n);
    I_p = I_H(n) + dt*beta*S_H(n)*I_H(n) - dt*gamma*I_H(n);
    R_p = R_H(n) + dt*gamma*I_H(n);
    
    S_H(n+1) = S_H(n) - (dt/2.0)*(beta*S_H(n)*I_H(n) +...
                                              beta*S_p*I_p);
    I_H(n+1) = I_H(n) +...
               (dt/2.0)*(beta*S_H(n)*I_H(n) - gamma*I_H(n) +...
                         beta*S_p*I_p - gamma*I_p);
    R_H(n+1) = R_H(n) + (dt/2.0)*(gamma*I_H(n) + gamma*I_p );
end

plot(t, S_FE, t, I_FE, t, R_FE, t, S_H, t, I_H, t, R_H);
legend('S (FE)', 'I (FE)', 'R (FE)', 'S (H)', 'I (H)', 'R (H)',...
                             'Location', 'northwest');
xlabel('hours');
print('tmp', '-dpdf');  print('tmp', '-dpng');

Filename: SIR_Heun.m.

Exercise 58: Use Odespy to solve a simple ODE

Solve $$ u' = -au + b,\quad u(0)=U_0,\quad t\in (0,T]$$ by the Odespy software. Let the problem parameters \( a \) and \( b \) be arguments to the right-hand side function that specifies the ODE to be solved. Plot the solution for the case when \( a=2 \), \( b=1 \), \( T=6/a \), and we use 100 time intervals in \( [0,T] \).

Solution.

import odespy
import numpy as np
import matplotlib.pyplot as plt

def f(u, t, a, b):
    return -a*u + b

# Problem parameters
a = 2
b = 1
U0 = 2
method = odespy.RK2

# Run Odespy solver
solver = method(f, f_args=[a, b])
solver.set_initial_condition(U0)
T = 6.0/a  # final simulation time
N_t = 100  # no of intervals
time_points = np.linspace(0, T, N_t+1)
u, t = solver.solve(time_points)

plt.plot(t, u)
plt.show

Filename: odespy_demo.m.

Exercise 59: Set up a Backward Euler scheme for oscillations

Write the ODE \( u'' + \omega^2u=0 \) as a system of two first-order ODEs and discretize these with backward differences as illustrated in Figure 38. The resulting method is referred to as a Backward Euler scheme. Identify the matrix and right-hand side of the linear system that has to be solved at each time level. Implement the method, either from scratch yourself or using Odespy (the name is odespy.BackwardEuler). Demonstrate that contrary to a Forward Euler scheme, the Backward Euler scheme leads to significant non-physical damping. The figure below shows that even with 60 time steps per period, the results after a few periods are useless:

Solution.

compare(odespy_methods=[odespy.EulerCromer, odespy.BackwardEuler],
        omega=2, X_0=2,number_of_periods=6,
        time_intervals_per_period=60)

Filename: osc_BE.m.

Exercise 60: Set up a Forward Euler scheme for nonlinear and damped oscillations

Derive a Forward Euler method for the ODE system (4.68)-(4.69). Compare the method with the Euler-Cromer scheme for the sliding friction problem from the section Spring-mass system with sliding friction:

1. Does the Forward Euler scheme give growing amplitudes?
2. Is the period of oscillation accurate?
3. What is the required time step size for the two methods to have visually coinciding curves?

Solution.

def sliding_friction():
    from numpy import tanh, sign

    f = lambda v: mu*m*g*sign(v)
    alpha = 60.0
    s = lambda u: k/alpha*tanh(alpha*u)
    #s = lambda u: k*u
    F = lambda t: 0

    g = 9.81
    mu = 0.4
    m = 1
    k = 1000

    U_0 = 0.1
    V_0 = 0

    T = 1
    dt = T/100.

    methods = [odespy.EulerCromer, odespy.ForwardEuler]
    compare(methods, f=f, s=s, F=F, m=1, U_0=U_0, V_0=V_0,
            T=T, dt=dt, start_of_plot=0)

Filename: osc_FE_general.m.

Exercise 61: Discretize an initial condition

Assume that the initial condition on \( u' \) is nonzero in the finite difference method from the section A finite difference method; undamped, linear case: \( u'(0)=V_0 \). Derive the special formula for \( u^1 \) in this case.

Solution.

Filename: ic_with_V_0.pdf.

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