Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 3 / a

Use the Backward Euler method in time and the prescribed central finite differences in space. With $j\in \mathbb{Z}$ and $m\ge 0$, the implicit finite difference method for the advection-diffusion equation isEquivalently, for $\nu =\Delta t/\Delta x$ and $\mu =\Delta t/(\Delta x{)}^2$, the new time level solvesThe spatial terms are evaluated at the new time level, making this an implicit scheme. On the whole line there are no finite-endpoint boundary conditions; in the next part its solution is chosen in the l2 sequence space.