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matlab/document.xml
matlab/output.xml
metadata/coreProperties.xml
metadata/mwcoreProperties.xml
metadata/mwcorePropertiesExtension.xml
metadata/mwcorePropertiesReleaseInfo.xml
Runge-Kutta Methods In this livescript, you will learn how To solve initial value problems using Runge-Kutta methods. Previously, we saw that with the explicit Euler method, we saw that the value of x_{n+1} is given by x_{n+1}=x_{n}+\Delta tf(t_{n},x_{n}) However, the only problem with it was that it had a low order of convergence \mathcal{O}(\Delta t) . We also saw that if we included n additional degrees of freedom by using the n higher order derivatives, we could increase this order of converegnce to \mathcal{O}(\Delta t)^{n+1} . But the problem remains that we have to evaluate these derivatives, which can be quite difficult. The main idea with Runge-Kutta methods is to estimate the derivative of x using a weighted average of the gradient between t_{n} and t_{n+1} g=a_{1}k_{1}+a_{2}k_{2}+\cdots+a_{N}k_{N} where the coefficients a_{i} and the gradients k_{i} are chosen in such a way that we can achieve the required order of convergence. For second order convergence, we have a_{1}=a_{2}=\frac{1}{2} k_{1}=f(t_{n},x_{n}) \\
k_{2}=f(t_{n}+\Delta t,x_{n}+k_{1}\Delta t) Then the value of x_{n+1} is given by x_{n+1}=x_{n}+\Delta tg The implementation of this in MATLAB is in fact quite simple as all we have to do is include additional lines of code to estimate the derivative. To illustrate this, consider the solution of the nonlinear problem \frac{dx}{dt}=-xt^{2} with the initial condition x(0)=1 . As always, we’ll initialise our variables and relevant parameters k = 1; %Growth rate
dt = 0.01; %Step size
t = 0:dt:10; %Time
x = zeros(size(t)); %Population size
x(1) = 1; %Initial condition
f = @(t,x) -x.*t.^2; %Forcing Function Then the main loop for the time stepping is given by for i = 1:length(t)-1
k1 = f(t(i),x(i));
k2 = f(t(i)+dt,x(i)+k1*dt);
g = 0.5*(k1+k2);
x(i+1) = x(i)+g*dt;
end
plot(t,x) Comparing with \texttt{ode45} we have [t1,x1] = ode45(f,[0,10],1);
hold on
plot(t1,x1)
hold off
legend(‘RK2′,’ode45’)
xlabel(‘t’)
ylabel(‘x’) (a) Estimate the maximum time-step such that the method is stable. (b) Modify the livescript to work with RK4.
manual code ready 0.4 figure 9be071f3-d6d5-41f9-8adc-7817a14bedbf 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 577 433 7 9 10 11 12 13 14 7 15 18 19 20 22 23 24 true false 0 21 21 false false 1 22 22 false false 2 23 23 false false 3 24 24 false false 4 25 25 false false 5 27 27 false false 6 29 34 false false 7 36 36 0 false false 8 38 38 false false 9 40 40 0 false false 10 41 41 0 false false 11 42 42 0 false false 12 44 44 0 false false 13 45 45 0 false true 14 46 46 0
2020-10-11T07:11:28Z 2020-10-18T05:40:36Z
application/vnd.mathworks.matlab.code MATLAB Code R2020b
9.9.0.1444674 f0a2ca84-b6b4-4982-aaa1-d965224f4793
9.9.0.1495850
R2020b
Update 1
Sep 30 2020
2395102982