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The usual way to define the implicit and explicit numerical (Finite difference) solution/discretization is by using a parabolic equation like the transient heat equation. How to define the implicit and explicit solution for a elliptic equation like the steady state heat conduction equation which just has spatial derivatives or is it even possible?(Is it even possible to talk about implicit and explicit schemes for steady state heat conduction equation?)
Also, what does an iterative method for an explicit scheme means as an explicit solution means as we can get the solution directly without any system of the equation (as the unknowns are given in terms of knows, and hence explicit)? Arent iterative methods used to solve the system of equations? Does it mean that the explicite solution also gives a system of equations or is it some other kind of iterative solution here or is it wrong to talk about an iterative solution for an explicit method?

GRANZER
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1 Answers1

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Simply put, explicit methods define $U_{n}$ in terms of the known values , $ U_{n-1},..., U_{n-k}$ while the implicit methods define $U_n$ in terms of $U_{n-1}, U_n, U_{n+1}$ where some of these values are not known.

Explicit methods are straight forward while implicit methods are more involved and time consuming to implement.

Mohammad Riazi-Kermani
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  • Sir, for the discretization of the parabolic equation, lets take here transient heat conduction equation, this makes sense as the solution at n+1 time given in terms of n-th time step values ( which are known) is called a explicit solution and the solution at n+1 time given in terms of n-th time step values ( which are known) and (n+1)th time step values(unknows) is called a implicit solution.... – GRANZER May 01 '19 at 06:00
  • ...How to do the same (differentiate b/w explicit and implicit form of solution) for elliptic equation(lets take Laplace equation, no temporal derivative) and what would be the known and unknown terms here? Also, what is the form of the usual five point solution for the elliptic equation?( is it explicit or implicit?) – GRANZER May 01 '19 at 06:00
  • @MohammadRiazi-Kermani In the case of implicit methods, I think you meant to say that $U_n$ is defined in terms of $U_n, U_{n-1}, \cdots, U_{n-k}$, right? – PierreCarre May 03 '19 at 08:54
  • You can also have $U_{n+1}$ is the implicit scheme. – Mohammad Riazi-Kermani May 03 '19 at 13:51