I understand what an implicit and explicit form of finite-difference (FD) discretization for the transient heat conduction equation means. But I am not able to understand if it is possible to categorize the discretization of steady state heat conduction equation (without a source term, i.e Laplace equation) as implicit or explicit scheme as well?(or is it just implicit form of discretization as we use iterative solvers?)

Edit: Whenever I try to understand the meaning of explicit an implicit solvers, the notes and other resources always explain them using a (transient) time-varying equation. According to such resources For a transient equation if the solution for the next time step is given in terms of current time step values (which we already know, either by initial condition or by previous calculation) it is called as an Explicit solution, and it could be solved for directly without forming a system of equation. If the solution for next time step value is given in terms of current time step values (which we know) and next time step values (which we don't know) it leads to an implicit solution and this gives us a system of equations which is solved for by iterative solvers.

My confusion is, if we take a steady state equation there are no time steps involved (so there is no current and next time step) so what would be an explicit and implicit solution? and why would an explicit solution require an iterative solver as they can be solved for directly?


1 Answer 1


A non-iterative solution for steady-state heat flow must be implicit. You are solving for the temperatures "everywhere" in the structure in a single "step", so every boundary condition must affect the entire temperature distribution in the equations you are solving.

Iterative solutions for steady-state conditions can be either implicit or explicit. You can think of each iteration as defining some sort of time increment, and calculating how the temperature distribution changes from the initial guess to the final steady state conditions. Implicit methods are likely to converge faster, but explicit methods will work and have some advantages - for example, it is much easier to program an implicit method on a massively parallel computer system like a GPU with hundreds or thousands of cores.

  • $\begingroup$ I was under the impression that non-iterative solvers were used with explicit solutions as there won't be any system of equations required(unknown solution variable is given in terms of known quantities.) and using iterative solvers means the solution is implicit (unknown is given in terms of unknown quantities which leads to system of equations). $\endgroup$
    Apr 29, 2019 at 18:07
  • $\begingroup$ Can you please explain the explicit and implicit solver ? $\endgroup$
    Apr 29, 2019 at 19:03

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