5
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ That bound is one way to state that the overall solution magnitude must either remain the same or decrease over time by enforcing that its time derivative is $\leq 0$. The high level result of this is analogous to other forms of numerical stability in ODEs where we don’t want our solution to grow unbounded over time, so we require that the real part of the eigenvalues for the (linearized) system should be $\leq 0$. So at least intuitively, what they list makes sense. The proof is just the fun part ;) $\endgroup$ – spektr Jun 29 at 23:33
11
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Thanks! In the book the PDE is denoted as wave equation. But actually the wave equation is $\Delta u = c u_{tt}$, meaning it is of second order, while this example is of first order. That is a bit strange to me. $\endgroup$ – dba Jun 30 at 11:24
  • 4
    $\begingroup$ @dba $u_t + c u_x = 0$ goes by many names: transport equation, first order wave equation, advection (or convection) equation, continuity equation... For constant speed $c$, it admits wave-like solutions of the form $u_0(x \pm ct)$, like its second-order counterpart - see here $\endgroup$ – GoHokies Jun 30 at 15:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.