Confusion about terminology : Finite difference for option pricing

Question

Consider the following initial-boundary value problem for $u = u(x,t),$

$$u_t - a u _{xx} = f(x,t) \text { for } 0 < x < L \text { and } 0 < t< T$$ along with bunch of initial and boundary conditions.

If we want to employ finite difference method to solve the above parabolic problem in 1D, then under explicit Euler method, we use forward difference approximation in time and under implicit Euler method, we use backward difference approximation in time.

I'm slighty confused about the terminology for finite difference method for option valuation. I was reading Paolo Brandimarte's Numerical Methods in Finance and in chapter 9 Option Pricing by finite difference method, to solve Black-Scholes PDE, under explicit scheme, he approximates the derivative with respect to time by a backward difference and under implicit scheme, he uses forward difference approximation in time. Isn't that the exact opposite of the previous definition?

Does it have something to do with terminal conditions in Black-Scholes PDE rather than initial condition and we go backward in time ?

Answer 1

An explicit (resp. implicit) finite difference scheme means you do not need to (resp. have to) solve a linear system of equations to find the solution at each intermediate time step. Whence their names!

Consequently, it really depends on if you have an initial, versus a terminal condition in time, and in what 'axes' you express and how you discretise the PDE you are solving.