semi-discrete method for second order PDE

Question

For the numerical method of second order PDE, we use semi-discrete(finite difference in $x$, continuous in $t$) method to translate PDE to a ODE system $$\dfrac{\textrm{d} U}{\textrm{d} t} = AU,$$ $$U(0) = U_0$$ here $$U(t) = (u_1(t),\dots,u_{J-1}(t))$$ is the vector of discrete numerical solution. $A$ is a constant coefficient matrix

My question is this ODE system can already be solved explicitly, why do we need the full-discrete method again i.e difference on time $t$ $$\dfrac{U^{n+1} - U^n}{k} = A U^{n+1},\quad 0\leq n\leq N-1$$ $$U^0 = U_0.$$ Here is $k = \Delta t.$

Answer 1

Speaking of numerical methods, even if you have an explicit solution of the ode in question, namely $$ U(t) = \exp(tA)U_0 $$ that does not really help. You have to compute $\exp(tA)$ somehow. As - for general matrices $A$ - computing an eigendecomposition is very badly conditioned, the usual way is a bad idea. You have to approximate $\exp$ by a rational function $R$, giving you $$ U_{n+1} = R(kA)U_n $$ $k = \Delta t$ as above. This is the very same thing as applying a RK-method to your equation.