3.2.3 Definitions

• Global error: difference between the true and computed solution

• Convergence: a FD scheme is convergent if the global error goes to zero as the mesh is refined

• Global order of accuracy: a method has global order $p$ if the global error is ${\cal O}(\Delta t^p + \Delta x^p)$ as $\Delta t, \Delta x \rightarrow 0$

• Local truncation error: measures how well the FD equations model the PDE locally. It is defined by inserting the true solution into the FD equations at a single arbitrary point.

  For smooth functions we can expand in Taylor series to obtain the local order of the method

  If the LTE vanishes like ${\cal O}(\Delta t^p + \Delta x^p)$ the method is (locally) $p$'th order accurate

  If the LTE goes to zero as the mesh is refined the method is called consistent

• Explicit and implicit methods: an explicit method is one which yields an explicit expression for each value $w_j^{n+1}$ at time $t^{n+1}$ in terms of nearby values at time $t^n$

  An implicit method couples together values at different grid points at time $t^{n+1}$ and an algebraic system of equations must be solved in each time step to update the solution

  Explicit methods are generally preferable provided they can be used with a reasonable time step (which depends upon grid resolution and smoothness of the solution). In many cases, an explicit method turns out to be unstable unless the time step is considerable smaller than what seems reasonable from accuracy considerations

• The CFL condition: a numerical method can be convergent only if its numerical domain of dependence contains the true domain of dependence of the PDE, at least in the limit as $\Delta t$ and $\Delta x$ go to zero

  For a hyperbolic system with eigenvalues $\lambda$ the CFL condition would imply $\vert\lambda\Delta t/\Delta x\vert \le 1$ as a necessary condition for stability