C++ 2D Wave Equation Solver

A 2D wave equation solver for a variable velocity model using the finite difference method.

Velocity model (top-left) and modelled pressure at increasing times. Reflections from the first velocity contrast can be seen from 4 seconds, and an emergent head-wave can be observed at 10-12 seconds.

Formulation

1. The wave equation

The 2D wave equation is defined as follows:

$$ \frac{\partial^2 u}{\partial t^2} = \gamma\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) $$

where $u$ is the pressure, $t$ is time, $\gamma=v^{2}$, and $v$ is the velocity of the medium that the wave is travelling through.

2. Discretization and finite differences

A numerical solution to the wave equation can be obtained using the finite difference method (FDM). In order to implement the FDM, the space and time domains are discretized as follows:

$$ x_{i} = i \Delta x\ \ \ \textup{for}\ i = 0, 1, 2,...,n_{x}-1, $$

$$ y_{j} = j \Delta y\ \ \ \textup{for}\ j = 0,1,2,...,n_{y}-1, $$

$$ t_{l} = l \Delta t\ \ \ \textup{for}\ l = 0,1,2,...,n_{t}-1. $$

The wave equation can then be expressed as

$$ \frac{\partial^2 u}{\partial t^2}=\frac{\partial }{\partial x}\left ( \gamma \frac{\partial u}{\partial x} \right )+\frac{\partial }{\partial y}\left ( \gamma \frac{\partial u }{\partial y} \right ). $$

The components on the right-hand side were approximated by finite differences in two stages:

1. The outer expression for each component was discretized as follows:

$$ \frac{\partial }{\partial x}\left ( \gamma \frac{\partial u}{\partial x} \right )\approx \frac{1}{\Delta x}\left ( \gamma\frac{\partial u}{\partial x}\bigg\rvert_{x=x_{i}+0.5} - \gamma\frac{\partial u}{\partial x}\bigg\rvert_{x=x_{i}-0.5} \right ). $$

2. The inner expressions were then approximated by the following:

$$ \begin{align*} \gamma\frac{\partial u}{\partial x}\bigg\rvert_{x=x_{i}+0.5}&\approx 0.5(\gamma_{i}+\gamma_{i+ 0.5})\frac{u_{i+1}+u_{i}}{\Delta x}, \\ \gamma\frac{\partial u}{\partial x}\bigg\rvert_{x=x_{i}-0.5}&\approx 0.5(\gamma_{i}+\gamma_{i- 0.5})\frac{u_{i}-u_{i-1}}{\Delta x} \end{align*} $$

The term on the left-hand side was then approximated by the following finite difference expression:

$$ \frac{\partial^2 u}{\partial t^2}\approx \frac{u_{ij}^{l-1}-2u_{ij}^{l}+u_{ij}^{l+1}}{\Delta t^{2}}. $$

Substituting the above expressions into wave equation and rearranging yields the following formula for calculating the pressure, $u_{ij}$, at the future time-step, $t=l+1$:

$$ u_{ij}^{l+1}=2u_{ij}^{l}-u_{ij}^{l-1}+\Psi_{ij} $$

where $\Psi_{ij}$ was defined as follows:

$$ \begin{align*} \Psi_{ij}&=\frac{\Delta t^{2}}{\Delta x}\left ( 0.5(\gamma_{ij}+\gamma_{i+1,j})\frac{u_{i+1,j}-u_{ij}}{\Delta x} - 0.5(\gamma_{ij}+\gamma_{i-1,j})\frac{u_{ij}-u_{i-1,j}}{\Delta x} \right )\\ &\quad +\frac{\Delta t^{2}}{\Delta y}\left ( 0.5(\gamma_{ij}+\gamma_{i,j+1})\frac{u_{i,j+1}-u_{ij}}{\Delta y} - 0.5(\gamma_{ij}+\gamma_{i,j-1})\frac{u_{ij}-u_{i,j-1}}{\Delta y} \right ). \end{align*} $$

This time-stepping expression is used to update the state of the inner points of the domain for each time increment.

3. Initial conditions

Initial conditions are required at $t=0$. Firstly, $u(t=0,x,y)=I(x,y)$ was used to define $u_{ij}^{l=0}$. $I(x,y)$ was chosen to be the Ricker wavelet function:

$$ I(x,y)=\frac{a}{\pi s^{2}}\left ( 1 - 0.5 \left ( \frac{x^{2}+y^{2}}{s^{2}} \right ) \right )e^{-\frac{x^{2}-y^{2}}{2s^{2}}} $$

where $a$ is the amplitude and $s$ is the spread. Secondly, $\frac{\partial u}{\partial t}(t=0,x,y)=0$ was approximated as follows:

$$ \frac{\partial u}{\partial t}\approx \frac{u_{ij}^{l+1}-u_{ij}^{l-1}}{2\Delta t}=0, $$

illustrating that $u_{ij}^{l+1}=u_{ij}^{l-1}$. Substituting $u_{ij}^{l+1}$ for $u_{ij}^{l-1}$ in the discretized wave equation, rearranging, and utilizing the fact that $u_{ij}^{l+1}=u_{ij}^{l-1}$, the following expression for the fictitious value of $u_{ij}^{l-1}$ at $t=0$ is obtained:

$$ u_{ij}^{l-1}=u_{ij}^{l}+\frac{\Psi_{ij}}{2} $$

4. Algorithm

The basic algorithm for the program can then be expressed as follows:

1. Set initial condition for all points: $u_{ij}^{l=0}=I(x,y)$
2. Set fictitious prior displacement for inner points: $u_{ij}^{l-1}=u_{ij}^{l}+\frac{\Psi_{ij}}{2}$
3. While $t\lt n_{t}$, update inner points: $u_{ij}^{l+1}=2u_{ij}^{l}-u_{ij}^{l-1}+\Psi_{ij}$
4. While $t\lt n_{t}$, initialise all points for the next time-step: $u_{ij}^{l-1}=u_{ij}^{l}$, $u_{ij}^{l}=u_{ij}^{l+1}$, $t=t+\Delta t$

Usage

Compile main.cpp and run the program:

g++ main.cpp -o wave-sim
./wave-sim

Parameters will be automatically prompted from the user, or they can be defined in runfile.txt where they are organised in the following order:

1. Size of the model in $x$ (lx)
2. Size of the model in $y$ (ly)
3. Spatial grid spacing in $x$ (dx)
4. Total time to run simulation (t)
5. Courant number (c)
6. Velocity of upper layer (v1)
7. Velocity of upper-mid layer (v2)
8. Velocity of lower-mid layer (v3)
9. Velocity of lower layer (v4)
10. Source amplitude (a)
11. Source spread (s)
12. Source location in $x$ given in cells (ox)
13. Source location in $x$ given in cells (oy)

The program writes a file named wave-sim.txt containing the 2D array, $u_{ij}$, at the final specified time, $t$.

Note that no dy is required as the model is assumed to be isotropic, and all parameters are given in SI units unless otherwise stated. The boundaries are reflecting ($u=0$), and the Courant number needs to be chosen carefully to avoid numerical dispersion and instability.

References

• Langtangen, H. P. (2013). A Primer on Scientific Programming with Python. Springer.