I have been thinking about numerical methods for ODEs, and the whole notion of stability confuses me. Take Euler's method for solving an ODE: U_n+1 = U_n + h.A.U_n where U_n = U_n( t ), A is the Jacobian and h is step size. Rearrange: U_n+1 = ( 1 + hA ).U_n This method is only stable if (1 + hA) < 1 ( using the eigenvalues of A). But what does this mean!?? Every value of my function that I am numerically getting is less than the previous value. This seems rather useless, I don't get it? It appears to me that this method can only be used on functions that are strictly decreasing for all increasing t ???