Can someone please tell me what discretization of a PDE is?
I was looking on youtube for a demonstration of the KDV equation:
which uses: Discretization, h_t = h_x^3
Can anyone explain this to me please?
When we are solving an ODE numerically, we don't get a continuous analytical solution - we only get solutions at discrete points.
ODEs are only a function of two variables, so in that case, you are only discretising the indepedent variable (for examples if you were solving y'' = f(y', y, t), then you would discretise the variable t into intervals of t, then evaluate the ODE at each point.
The same is true of PDEs except that PDEs have more than one independent variable. Most usually, the PDEs have two independent variables, so you have to discretise 2 of these variables into a 2D gride of node points. You then evaluate the solution of the PDE at each node point.
No problem. The discretisation basically describes the shape of his grid.
The two independent variables he has chosen to discretise are t and x, and because it is a 2D grid you are discretising, you need to choose the shape of the grid. His grid shape is defined by that equation. Although in most cases it's satisfactory to have just a two straight line parallel grids. Hope this helps.