The Backward Euler scheme:
un+1=un+Δt(aun+1+bvn+1),vn+1=vn+Δt(cun+1+dvn+1). which is a 2×2 linear system:
(1−Δta)un+1+bvn+1=un,cun+1+(1−Δtd)vn+1=vn,
Crank-Nicolson also gives a 2×2 linear system.