# Can you describe the Galerkin numerical method to solve the wave equation?

How would you describe the Galerkin method to solving the 3D wave equation $$u_{tt}= c^2\Delta u$$ to someone who wants to implement it immediately?

More precisely, we want to solve the Cauchy problem $$u_{tt}= c^2\Delta u , \\ u_t(x,y,z,0)= g(x,y,z) , \\ u(x,y,z,0)= f(x,y,z)$$ where $$f, g$$ are given compactly supported functions.

Say in a bounded region of the space we are interested in, say a ball of radius $$R$$, note that the wave propagates towards the boundary of the region, the sphere, thus we can't impose the condition at the boundary.

How would you implement this method ? I am looking for a description of the finite elements, how to decompose this region? (you can use any other shape, not necessarily a ball, take a cube for example), then describing the numerical algorithm necessary to find the values of the solution $$u$$ at certain element or node $$(x,y,z,T)$$ at time $$T$$

• This is one of the three standard equations discussed in nearly every book about the finite element method. Have you taken a look at the typical finite element books? May 19 at 17:24
• What book would you recommend please? May 20 at 2:03
• I doubt this is a Cauchy problem since you don't have boundary conditions. May 20 at 17:21