I am doing a research to explore the existing numerical schemes that are used to solve the $3$D wave equation.
The standard form of the problem in $3$ dimensional setting is : $$\Delta u= \frac{1}{c^2} u_{tt}\ ,\ c\text{ is constant}$$
and $ u: \mathbb{R}^3\times \mathbb{R}\rightarrow \mathbb{R} $ , with initial conditions :
$u(x,0)= f(x)\ ,\ u_t(x,0)= g(x)$ , for arbitrary density functions $f, g: \mathbb{R}^3\rightarrow \mathbb{R}$ (though generally compactly supported)
I am developing a new scheme that should be practical in many settings, now I need to compare it to other schemes to check what is the most efficient.
Are schemes based on the finite elements method the only schemes used ? or is there anything else?
For constant $c$, you can also transform it into a system of hyperbolic equations: Define $$ v := c \nabla u, \quad w := -u_t$$ Then, \begin{align} w_t +c \nabla \cdot v &= 0 \\ v_t + c \nabla w &= 0 \end{align} Where the first equation is the original PDE, and the second equation is needed to close the system (although it is fulfilled by definition). Initial and boundary conditions follow this definition.
In this form, you would typically employ finite volume methods instead of finite element methods.