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In the book of Hesthaven and Warburton on discontinuous Galerlkin methods in example 2.3 (regarding solutions of the wave equation) the authors regard the following pde:

$$fracpartial u partial t + afracpartial upartial x = 0,
x in [L,R] = Omega
$$
They state that for stability of the numerical scheme, the following must hold:

$$sum_k=1^K fracddt||u_h^k||^2_D ^k=fracddt||u_h^k||^2_Omega,h leq 0 \ bigcup_k^K D^k = Omega$$

with $D^k$ nonoverlapping intervals.
Unfortunately this comes without a proof and I dont have the intuition to see why this may be true. I assume that stability means that small changes in the initial data should only lead to small changes in the solution.

Stability does indeed mean that small changes in the data lead to small changes in the solution. This can be shown for the linear advection equation through the energy method; in proving the stability of a discretization, we seek to mimic the energy estimate of the exact problem. Considering the PDE,
$$
fracpartial upartial t + a fracpartial upartial x = 0, quad x in Omega equiv (x_L, x_R), quad t in Iequiv(0,T), quad a > 0,
$$
subject to the initial condition
$
u(x, t=0) = u_0(x)
$
and the inflow boundary condition
$
u(x =x_L, t) = u_L(t),
$
we can apply the energy method by multiplying the PDE by $u$ and integrating over $Omega$:
$$
int_Omega u fracpartial upartial t ,mathrmdx + aint_Omega u fracpartial upartial x ,mathrmdx = 0.
$$
By the chain rule, we note that
$$
u fracpartial upartial t = frac12fracpartialpartial tleft(u^2right),
$$
and applying integration by parts,
$$
int_Omega u fracpartial upartial x ,mathrmdx = frac12u^2Big|_x_L^x_R.
$$
Therefore,
$$
int_Omegafracpartialpartial tleft(u^2right) ,mathrmdx = -au^2Big|_x_L^x_R = -aleft[u(x=x_R, t)right]^2 + aleft[u_L(t)right]^2.
$$
Applying Leibniz's rule on the left-hand side and noting that $-aleft[u(x=x_R, t)right]^2 leq 0$,
$$
fracmathrmdmathrmdt||u(cdot,t)||_L^2(Omega)^2 leq aleft[u_L(t)right]^2.
$$
Integrating in time gives us
$$
||u(cdot,T)||_L^2(Omega)^2 - ||u_0||_L^2(Omega)^2 leq a int_I [u_L(t)]^2,mathrmdt,
$$
so the solution is bounded in terms of the problem data (the initial and boundary conditions) as
$$
||u(cdot,T)||_L^2(Omega)^2 leq ||u_0||_L^2(Omega)^2 + a int_I [u_L(t)]^2,mathrmdt,
$$
which corresponds to your notion of stability. In the homogeneous case where $u_L = 0$, we recover
$$
fracmathrmdmathrmdt||u(cdot,t)||_L^2(Omega)^2 leq 0,
$$
which (through integration in time) implies that
$$
||u(cdot,T)||_L^2(Omega)^2 leq ||u_0||_L^2(Omega)^2.
$$
The discontinuous Galerkin method mimics such an energy estimate for the linear advection equation (as does any scheme which satisfies a generalized summation-by-parts property, provided that interface and boundary conditions are treated appropriately). It is common to assume a homogeneous inflow boundary condition and simply show that the energy is nonincreasing with time; however, as we have seen, the motivation for doing so is to bound the solution in terms of the data of the problem.