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$\newcommand{\pd}[2]{\frac{\partial#1}{\partial#2}} \newcommand{\IR}{\Bbb R}\newcommand{\dd}{\mathrm{d}}$ The wave equation: $$ \pd{^2u}{t^2}(x,t)=c^2\pd{^2u}{x^2}(x,t),~~ t>0,~x\in(0,1)\tag4 $$ Given wave equation with start and end values

where $c$ is constant (sound / light speed), can be approximated in two steps:

  • define the vector $u(t) ∈ \IR^{N + 1}$, $u(0,t) = u(1,t) = 0$, where $\bar u_j(t) ≃ u(j/N, t)$ for $j = 0,1,2, ..., N$ and let $$ N^2 (\bar u_{j+1}(t) −2\bar u_{j}(t) + \bar u_{j-1}(t) ) ≃ \pd{^2 u(j / N, t)}{x^2} $$ be the usual approximation of the second derivative. Then we get the following system of differential equations (image 2): $$ \frac{\dd^2\bar u_j(t)}{\dd t^2}=c^2N^2 (\bar u_{j+1}(t) −2\bar u_{j}(t) + \bar u_{j-1}(t) ), ~~ j=1,...,N-1,\tag5 $$ Image 2

so that $\frac{\dd^2 \hat u(t)}{\dd t^2} = c^2 A \hat u(t)$ where $u(t) ∈ \IR^{N − 1}$ is the vector with the components $\hat u_j(t)=\bar u_j(t)$, $j = 1,..., N-1$ vector kompenents expression and $A$ are the $(N −1) × (N −1)$ matrix with $−2N^2$ in the diagonal and $N^2$ in the upper and lower diagonals.

Show that the energy $(|\frac{\dd\hat u(t)}{\dd t}|^2)/2 - c^2\hat u(t) · A\hat u(t)/2$ is constant for all times, where "$·$" means $N-1$ scalar product in $\IR^{N − 1}$. Suggest a suitable numerical method.

Karl Karlsson
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1 Answers1

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This is a second order equation of the form $\ddot x=f(x)$. You can apply any method that is developed especially for this case like Numerov's method1, Beeman's method or more modern methods that are found in molecular dynamics simulations.

Given that you have an energy function that can serve as Hamiltonian of the equation, you can also use any symplectic method, most famously the Verlet second order method, its 4th order modification or the Ruth methods.

Or you could transform the second order system into a first order system of twice the size and apply any ODE solver a là method-of-lines.

Lutz Lehmann
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