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4

Ok, here comes the answer promised in the comment section. Let's start the other way round, going from a general grid to Gaussian grids and further constructions such as spectral elements. In grid methods, one basically selects a number of $N+1$ gridpoints $\{x_k\}_{k=0}^{N}$. As basis functions, one can use Lagrange polynomials constructed over these nodes, ...

3

$c$ depending on time is not the issue. You will use an RK scheme which takes care of this. The issue is $c$ is discontinuous in $x$. I recommend SBP-SAT schemes for this. (1) Derive an energy equation at PDE level. (2) Search literature for SBP-SAT schemes which enforce interface conditions via SAT penalty terms, which are designed to mimic the energy ...

2

Let's consider the one-dimensional string first. Standard text-book physics considers the three usual boundary conditions here, namely Dirichlet (endpoints of the string are fixed), Neumann (endpoints are free) and Robin conditions (obtained e.g. when the endpoints are attached to a spring). Now, for real sonic propagation, those boundary conditions won't ...

2

Here is a brute force solution that would work no matter what is the discontinuity and nonlinearity in $c(x,t)$. Write your PDE as a system of two: $\dot{y}=z\\ \dot{z}=c^2(x,t) y_{xx}$ Now, discretize it on a uniform spatial grid in x: $\vec{x}= [x_0, x_1,..., x_{n-1}] \\ \vec{y}= [y_0, y_1,..., y_{n-1}] \\ \vec{z}= [z_0, z_1,..., z_{n-1}] \\$ Now the ...

2

As David said, absorbing boundary conditions won't be completely reflectionless. That said, we can reduce relfections quite a bit, which helps to avoid influence from the boundaries while the particle still travelling inside. Since this is a time dependent problem, one simple choice of boundary conditions will look like this. At the left boundary: \frac{\...

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