MTH3007B Weekly Problems 2

Original Documents: Problem Sheet

Vibes: Pretty chill, basically just the same as last week and scaling errors.

Used Techniques:

• Scaling errors.
• Implementing Euler methods.

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2.1. Scaling Error in Different Methods

Question

Assume you solve a differential equation, $y^{\prime }(t)=g(t,y(t))$, using the Explicit Euler method using a step size of $\Delta t=0.01$. Also assume the (global) error is exactly as in the linear regime and the error in $y(t_{\text{max}})$ is $0.04$.

1. What would the error be if $\Delta t=0.005$?
2. What would the error be if $\Delta t=0.005$ and Ralston is used, instead? Assume an error of $0.03$ for $\Delta t=0.01$.

The global error scales with the timestep according to the order of the method…

1. For Forward Euler, a first order method, then $E\propto \Delta t$. Hence, halving the step size halves the error: $E_{\text{new}}=E_{\text{old}}\cdot \frac{\Delta t_{\text{new}}}{\Delta t_{\text{old}}}=0.04\times \frac{0.005}{0.01}=\boxed{0.02}$.
2. For Ralston, a second order method, then $E\propto \Delta t^2$. Hence, halving the step size quarters the error: $E_{\text{new}}=E_{\text{old}}\cdot {\left(\frac{\Delta t_{\text{new}}}{\Delta t_{\text{old}}}\right)}^2=0.03\times {\left(\frac{0.005}{0.01}\right)}^2=0.03\times 0.25=\boxed{0.0075}$.

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2.2. Numerical Solutions of an ODE

Question

Consider the ordinary differential equation, $\frac{dy(t)}{dt}=bt-ay(t)$ with $a=8$, $b=1$, and $y(0)=4$.

1. By using a numerical algorithm, solve the ODE till $t_{\max }=1$, then compare $y(t_{\max })$ with the analytical solution $y(t)=\exp (-at)\left(y(0)+b/a^2\right)+bt/a-b/a^2$.
2. What is $y(t_{\max })$ for the Ralston method for $\Delta t=0.01$?

1. Using the Explicit Euler method, the solution is $\boxed{0.1103355852}$. Analytically, the solution is approximately $0.1107220921$ - a difference of about $3.865\times 10^{-4}$
2. Instead, using the Ralston method, the solution is instead $\boxed{0.1107343545}$, giving a smaller max error of $1.226\times 10^{-5}$.

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2.3. Finding Maximum Timestep for a Given Error in Euler

Question

Assume you want to have an accuracy in the final solution of $0.001$, such that the numerical result of $y(t_{\text{max}})$ can deviate at most $0.001$ from the analytical solution.

What is the maximum timestep $\Delta t$ for the Explicit Euler method to achieve the $0.001$ accuracy in $y(t_{\text{max}})$, with one significant figure accuracy?

Using the linear error scaling for Euler with the known error at $\Delta t=0.01$…

$$\Delta t_{\text{max}}=\Delta t\cdot \frac{0.001}{E(\Delta t)}=0.01\times \frac{0.001}{3.865\times 10^{-4}}\approx \boxed{0.03}$$

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2.4. Finding Maximum Timestep for a Given Error in Ralston

Question

What is the maximum timestep $\Delta t$ for Ralston’s method to achieve the $0.001$ accuracy in $y(t_{\text{max}})$, with one significant figure accuracy?

Using the quadratic error scaling of Ralston with the known error at $\Delta t=0.01$…

$$\Delta t_{\text{max}}=\Delta t\cdot \sqrt{\frac{0.001}{E(\Delta t)}}=0.01\times \sqrt{\frac{0.001}{1.226\times 10^{-5}}}\approx \boxed{0.09}$$

This is apparently incorrect - naive scaling?