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</style></head><body><div class = "content"><div class = 'SectionBlock containment'><h1 class = "S1"><span class = "S2">Exercise 6: Non-linear problems</span></h1><h2 class = "S3"><span class = "S2">Illustrative example</span></h2><div class = "S4"><span class = "S2">Consider the differential equation describing the motion of frictionless pendulum</span></div><div class = "S5"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="165" height="40" /></span></div><div class = "S4"><span class = "S2">where </span><span style="font-family:Symbol, 'DejaVu Serif', serif; font-style: italic; font-weight: normal">θ</span><span class = "S2"> is the angle, </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">g</span><span class = "S2"> is acceleration of gravity and </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">l</span><span class = "S2"> is the length. Since </span><span style="font-family:Symbol, 'DejaVu Serif', serif; font-style: italic; font-weight: normal">θ</span><span class = "S2"> depends only on </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">t</span><span class = "S2">, (1) is an initial value problem. We can convert it into a system of first order ODEs and solve it numerically with the methods we have seen during the course. Introducing the angular velocity </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="49" height="37" /></span><span class = "S2"> the first order system looks like</span></div><div class = "S5"><span style="vertical-align:-24px"><img src="data:image/png;base64,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" width="218" height="58" /></span></div><div class = "S4"><span class = "S2">Defining </span></div><div class = "S5"><span style="vertical-align:-24px"><img src="data:image/png;base64,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" width="243" height="58" /></span></div><div class = "S4"><span class = "S2">we can discretize the problem with implicit Euler scheme as</span></div><div class = "S5"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="202.5" height="53" /></span></div><div class = "S4"><span class = "S2">and solve it with either fixed-point algorithm or Newton method.</span></div></div><div class = "S0"></div><div class = 'SectionBlock containment active'><h2 class = "S6"><span class = "S2">Fixed-point algorithm</span></h2><div class = "S4"><span class = "S2">For better readability we define </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="73.5" height="32" /></span><span class = "S2"> and rearrange (3) to the fixed-point form as</span></div><div class = "S5"><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="130.5" height="33" /></span></div><div class = "S4"><span class = "S2">The MatLab code below solves (3) with the fixed-point algorithm.</span></div><div class = 'LineNodeBlock contiguous'><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Parameters</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">l=1;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">g=9.81;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Initial condition</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">U = [ pi/4 ; 0 ];</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Function f</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">f = @(U) [ U(2) ; -g/l*sin(U(1)) ]; </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Options for time-stepping</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">dt = 0.01;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">tmax = 4;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Options for non-linear solver</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">itmax = 1000;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">tol = 1e-5;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11">relaxation = 0.1</span><span class = "S10">;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Initialization</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">t = 0;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">V = U;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">K = round((tmax-t)/dt);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">theta = NaN(K,1);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">omega = theta;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">tic;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Time-stepping loop</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S12">for </span><span class = "S10">k=1:K</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S9">% Fixed-point algorithm</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> error = inf;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> it = 0;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S12">while </span><span class = "S10">error>tol && itmax>it</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> Vold = V;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> Vnew = U + dt*f(V);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> V = relaxation*Vnew + (1-relaxation)*V;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> it = it+1;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> error = norm(V-Vold);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S13">end</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S12">if </span><span class = "S11">it==itmax, disp(</span><span class = "S14">'Not converged'</span><span class = "S11">); </span><span class = "S13">end</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S9">% Proceed to next time-step</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> U = V;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> t = t + dt;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S9">% Save trajectory</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> theta(k) = U(1);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> omega(k) = U(2);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S13">end</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S13"></span></div></div><div class = 'inlineWrapper outputs'><div class = "S8 lineNode"><span class = "S11">ComputationTime</span><span class = "S10"> = toc</span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><div class="inlineElement embeddedOutputsVariableElement" uid="6F5EF64E" data-testid="output_0" style="width: 1171px; white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">ComputationTime = 9.9517</div></div></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S13"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Plot phase diagram</span></div></div><div class = 'inlineWrapper outputs'><div class = "S8 lineNode"><span class = "S10">plot(theta,omega);</span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><div class="inlineElement embeddedOutputsFigure" uid="0A2C46C3" data-testid="output_1" style="width: 1171px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><div class="figureElement" style="cursor: default; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><img class="figureImage" 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" style="width: 560px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"></div></div></div></div></div><div class = "S16"><span class = "S2">Note how the implicit Euler scheme dissipates energy, although the original equation models a frictionless pendulum.</span></div><div class = "S4"><span class = "S2"></span></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h2 class = "S6"><span class = "S2">Newton-Raphson method</span></h2><div class = "S4"><span class = "S2">We can also re-arrange (3) into the form</span></div><div class = "S5"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="191.5" height="51" /></span></div><div class = "S4"><span class = "S2">and approximate the roots of the non-linear function </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="15.5" height="30" /></span><span class = "S2"> at each time step by Newton-Raphson method. See the MatLab script below:</span></div><div class = 'LineNodeBlock contiguous'><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Re-initialization</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">U = [ pi/4 ; 0 ];</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">t = 0;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">theta = NaN(K,1);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">omega = theta;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">V = U;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Define F and J</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">I = eye(2);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">O = zeros(2);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">F = @(V,U) V - U - dt*f(V) ;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">J = @(V) I - O - dt*[ 0, 1; -g/l*cos(V(1)), 0 ];</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Time-stepping loop</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10">tic;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S12">for </span><span class = "S10">k=1:K</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S9">% Newton algorithm</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> error = inf;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> it = 0;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S12">while </span><span class = "S10">error>tol && itmax>it</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> deltaV = -J(V)\F(V,U);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> V = V + deltaV;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> error = norm(deltaV);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> it = it+1;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S13">end</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S12">if </span><span class = "S11">it==itmax, disp(</span><span class = "S14">'Not converged'</span><span class = "S11">); </span><span class = "S13">end</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S9">% Proceed to next time-step</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> U = V;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> t = t + dt;</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S11"> </span><span class = "S9">% Save trajectory</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> theta(k) = U(1);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> omega(k) = U(2);</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S10"> </span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S13">end</span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S13"></span></div></div><div class = 'inlineWrapper outputs'><div class = "S8 lineNode"><span class = "S10">ComputationTime = toc</span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><div class="inlineElement embeddedOutputsVariableElement" uid="A7C940C7" data-testid="output_2" style="width: 1171px; white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">ComputationTime = 0.5604</div></div></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S13"></span></div></div><div class = 'inlineWrapper'><div class = "S8 lineNode"><span class = "S9">% Plot phase diagram</span></div></div><div class = 'inlineWrapper outputs'><div class = "S8 lineNode"><span class = "S10">plot(theta,omega);</span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><div class="inlineElement embeddedOutputsFigure" uid="96CA7240" data-testid="output_3" style="width: 1171px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><div class="figureElement" style="cursor: default; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><img class="figureImage" 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" style="width: 560px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"></div></div></div></div></div><h2 class = "S3"><span class = "S2"></span></h2></div></div>
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##### SOURCE BEGIN #####
%% Exercise 6: Non-linear problems
%% Illustrative example
% Consider the differential equation describing the motion of frictionless pendulum
%
% $$\frac{d^2 \theta}{d t^2} = -\frac{g}{l} \sin(\theta) \quad \quad \text{(1)}$$
%
% where $\theta$ is the angle, $g$ is acceleration of gravity and $l$ is
% the length. Since $\theta$ depends only on $t$, (1) is an initial value problem.
% We can convert it into a system of first order ODEs and solve it numerically
% with the methods we have seen during the course. Introducing the angular velocity
% $\omega = \frac{d \theta}{d t}$ the first order system looks like
%
% $$\frac{d}{dt}\left( \begin{array}{c}\theta \\ \omega\end{array} \right)=\left(
% \begin{array}{c}\omega \\ -\frac{g}{l} \sin(\theta)\end{array} \right)\quad
% \quad \text{(2)}$$
%
% Defining
%
% $$\vec{U} := \left( \begin{array}{c}\theta \\ \omega\end{array} \right),
% \quad\vec{f}(\vec{U}) := \left( \begin{array}{c}\omega \\ -\frac{g}{l} \sin(\theta)\end{array}
% \right)$$
%
% we can discretize the problem with implicit Euler scheme as
%
% $$\frac{\vec{U}^{n+1} - \vec{U}^n} {\Delta t}=\vec{f}(\vec{U}^{n+1})\quad
% \quad \text{(3)}$$
%
% and solve it with either fixed-point algorithm or Newton method.
%% Fixed-point algorithm
% For better readability we define $\vec{V} := \vec{U^{n+1}}$ and rearrange
% (3) to the fixed-point form as
%
% $$\vec{V} = \vec{U^n} + \Delta t \ \vec{f}(\vec{V})$$
%
% The MatLab code below solves (3) with the fixed-point algorithm.
%%
% Parameters
l=1;
g=9.81;
% Initial condition
U = [ pi/4 ; 0 ];
% Function f
f = @(U) [ U(2) ; -g/l*sin(U(1)) ];
% Options for time-stepping
dt = 0.01;
tmax = 4;
% Options for non-linear solver
itmax = 1000;
tol = 1e-5;
relaxation = 0.1;
% Initialization
t = 0;
V = U;
K = round((tmax-t)/dt);
theta = NaN(K,1);
omega = theta;
tic;
% Time-stepping loop
for k=1:K
% Fixed-point algorithm
error = inf;
it = 0;
while error>tol && itmax>it
Vold = V;
Vnew = U + dt*f(V);
V = relaxation*Vnew + (1-relaxation)*V;
it = it+1;
error = norm(V-Vold);
end
if it==itmax, disp('Not converged'); end
% Proceed to next time-step
U = V;
t = t + dt;
% Save trajectory
theta(k) = U(1);
omega(k) = U(2);
end
ComputationTime = toc
% Plot phase diagram
plot(theta,omega);
%%
% Note how the implicit Euler scheme dissipates energy, although the original
% equation models a frictionless pendulum.
%
%
%% Newton-Raphson method
% We can also re-arrange (3) into the form
%
% $$\vec{F}(\vec{V}) := \frac{\vec{V} - \vec{U^n}}{\Delta t} - \vec{f}(\vec{V})
% \overset{!}{=} 0$$
%
% and approximate the roots of the non-linear function $\vec{F}$ at each
% time step by Newton-Raphson method. See the MatLab script below:
%%
% Re-initialization
U = [ pi/4 ; 0 ];
t = 0;
theta = NaN(K,1);
omega = theta;
V = U;
% Define F and J
I = eye(2);
O = zeros(2);
F = @(V,U) V - U - dt*f(V) ;
J = @(V) I - O - dt*[ 0, 1; -g/l*cos(V(1)), 0 ];
% Time-stepping loop
tic;
for k=1:K
% Newton algorithm
error = inf;
it = 0;
while error>tol && itmax>it
deltaV = -J(V)\F(V,U);
V = V + deltaV;
error = norm(deltaV);
it = it+1;
end
if it==itmax, disp('Not converged'); end
% Proceed to next time-step
U = V;
t = t + dt;
% Save trajectory
theta(k) = U(1);
omega(k) = U(2);
end
ComputationTime = toc
% Plot phase diagram
plot(theta,omega);
%%
##### SOURCE END #####
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