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I am trying to answer a question about a finite difference scheme. I need to show that the method is stable in the L1-norm. I can't find a single definition of what that means, so does anyone have a source or know for a fact what the definition is?

I'm guessing that if we denote the operator corresponding to the scheme as $H$, then we need to show that $$||H^n||_1 \le C$$ where $C>0$ is a constant.

Thanks.

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  • $\begingroup$ In what context does the problem arise? In the numerical solution of PDEs? $\endgroup$ – ekkilop Aug 12 '16 at 9:22
  • $\begingroup$ Forgot to mention that in the body, just had it in the tag. It's for numerically solving PDEs with finite differencing. $\endgroup$ – Kurt Aug 12 '16 at 15:24
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Consider a general PDE of the form \begin{align} u_t &= P(x,t,\partial_x)u + F(x,t), \quad a \leq x \leq b, \quad t \geq 0 \\ u(x,0) &= f(x) \\ L_a(t,\partial_x) u(a,t) &= g_a(t) \\ L_b(t,\partial_x) u(b,t) &= g_b(t) \end{align} where $P$ is a differential operator with smooth matrix coefficients, $F$ is a forcing function, and $L_a$ and $L_b$ are differential operators defining the boundary conditions. Let this PDE be discretised in space by a finite difference scheme as follows: \begin{align} (v_j)_t &= Q_j(x_j,t) v_j + F_j + S_j, \quad j=0,\dots,N, \quad t \geq 0 \\ v_j(0) &= f_j \end{align} where $\mathbf{v}=(v_0, \dots, v_N)^T$ is the grid vector approximating $u$, $Q_j$ is the approximation of $P$ at $x_j$, and $S_j$ contains whatever modification you need of your difference scheme to incorporate the boundary conditions. $\mathbf{f} = (f_0, \dots, f_N)^T$ is the projection of $f$ onto the grid and similarly we define $\mathbf{F}$.

Now, consider the PDE above with $F=g_a=g_b=0$. The finite difference scheme is said to be stable in the norm $\| \cdot \|$ if $$ \| \mathbf{v}(t) \| \leq K e^{\alpha t} \| \mathbf{f} \| $$ holds, and $K$ and $\alpha$ are independent of $\mathbf{f}$.

For general forcing and boundary conditions, the finite difference scheme is said to be strongly stable in the norm $\| \cdot \|$ if, instead of the above, $$ \| \mathbf{v}(t) \|^2 \leq K(t) \left( \| \mathbf{f} \|^2 + \max_{\tau \in [0,t]} \| \mathbf{F}(\tau) \|^2 + \max_{\tau \in [0,t]} \| g_a(\tau) \|^2 + \max_{\tau \in [0,t]} \| g_b(\tau) \|^2 \right) $$ holds, and $K(t)$ is bounded for any finite $t$ and independent of $\mathbf{F},g_a,g_b$ and $\mathbf{f}$.

A possible way to obtain such estimates is through the energy method, which is outlined in most text books on the subject. This typically gives estimates in the 2-norm, which may possibly be extended to the 1-norm with the aid of various inequalities.

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