Verification via manufactured solutions

• Choose any formula for $u(t)$.
• Fit $I$, $a(t)$, and $b(t)$ in $u'=-au+b$, $u(0)=I$, to make the chosen formula a solution of the ODE problem.
• Then we can always have an analytical solution (!).
• Ideal for verification: testing convergence rates.
• Called the method of manufactured solutions (MMS)
• Special case: $u$ linear in $t$, because all sound numerical methods will reproduce a linear $u$ exactly (machine precision).
• $u(t) = ct + d$. $u(0)=0$ means $d=I$.
• ODE implies $c = -a(t)u + b(t)$.
• Choose $a(t)$ and $c$, and set $b(t) = c + a(t)(ct + I)$.
• Any error in the formula for $u^{n+1}$ makes $u\neq ct+I$!