Stability of hyperbolic PDE and DG-FEM

Question

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:

$$\frac{\partial u }{\partial t} + a\frac{\partial u}{\partial x} = 0, x \in [L,R] = \Omega $$ They state that for stability of the numerical scheme, the following must hold:

$$\sum_{k=1}^K \frac{d}{dt}||u_h^k||^2_{D ^k}=\frac{d}{dt}||u_h^k||^2_{\Omega,h} \leq 0 \\ \bigcup_k^K D^k = \Omega$$

with $D^k$ nonoverlapping intervals.

Unfortunately this comes without a proof, and I don't 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.

Answer 1

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, $$ \frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0, \quad x \in \Omega \equiv (x_L, x_R), \quad t \in I\equiv(0,T), \quad a > 0, $$ 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 $\Omega$: $$ \int_\Omega u \frac{\partial u}{\partial t} \,\mathrm{d}x + a\int_\Omega u \frac{\partial u}{\partial x} \,\mathrm{d}x = 0. $$ By the chain rule, we note that $$ u \frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial}{\partial t}\left(u^2\right), $$ and applying integration by parts, $$ \int_\Omega u \frac{\partial u}{\partial x} \,\mathrm{d}x = \frac{1}{2}u^2\Big|_{x_L}^{x_R}. $$ Therefore, $$ \int_\Omega\frac{\partial}{\partial t}\left(u^2\right) \,\mathrm{d}x = -au^2\Big|_{x_L}^{x_R} = -a\left[u(x=x_R, t)\right]^2 + a\left[u_L(t)\right]^2. $$ Applying Leibniz's rule on the left-hand side and noting that $-a\left[u(x=x_R, t)\right]^2 \leq 0$, $$ \frac{\mathrm{d}}{\mathrm{d}t}||u(\cdot,t)||_{L^2(\Omega)}^2 \leq a\left[u_L(t)\right]^2. $$ Integrating in time gives us $$ ||u(\cdot,T)||_{L^2(\Omega)}^2 - ||u_0||_{L^2(\Omega)}^2 \leq a \int_{I} [u_L(t)]^2\,\mathrm{d}t, $$ so the solution is bounded in terms of the problem data (the initial and boundary conditions) as $$ ||u(\cdot,T)||_{L^2(\Omega)}^2 \leq ||u_0||_{L^2(\Omega)}^2 + a \int_{I} [u_L(t)]^2\,\mathrm{d}t, $$ which corresponds to your notion of stability. In the homogeneous case where $u_L = 0$, we recover $$ \frac{\mathrm{d}}{\mathrm{d}t}||u(\cdot,t)||_{L^2(\Omega)}^2 \leq 0, $$ 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.