Mat1062: Introductory Numerical Methods for PDE

This notation is slightly confusing, since L is not a matrix: we use it to denote this \a ne linear" operation. To solve this system of ODEs, we choose a time step k, de ne time levels tn = nk for n = 0, 1, . . . , and introduce u n ∈ RN−1. The vector un will be determined by a discrete dynamical system and it is our hope that this can be done in such a way that uj ≈ Uj(tn) ≈ u(xj, tn). Recall that the superscript n denotes the nth time level rather than the nth power.