an explicit scheme is one where there is a simple updating procedure that does not depend upon other values at the current level while an implicit one contains information at the current level which requires the solving of simultaneous equations. The main pros and cons of these two methods are that explicit schemes are easy to program with only simple calculations performed at each timestep while implicit schemes are more difficult and can involve many iterations per timestep. Explicit schemes however have a tendancy to be unstable and require very small timesteps while implicit schemes are very stable and can have much larger timesteps.

Explicit schemes have conditional stability, i.e. that the size of time step in the case of heat equation is proportional to to square of spatial step. This is usually too restricting (you need many "iterations"). Therefore you may want to use some kind of implicit schemes, which is unconditionally stable (independent choice of time step with respect to spatial step). 

Dear Gayan,

in addition to the answers already given I would suggest two books in particular.

Richtmyer & Morton, Difference methods for initial-value problems, Interscience 1967.

Although this seems from a rather distant past Richtmyer-Morton is still a classic. On p.185-217 you will find what you need.

Alternatively, a very good book is Computational Physics by David Potter, Wiley & Sons1972 This book, written by a physicist, makes it clear that computational physics really is a combination of numerical math and physics. You need both disciplines to make sensible choices. I enjoyed it tremendously. You will find important information for your problem on p 76-79 and on p. 203 in case you want to combine heat flow with fluid dynamics.

I suppose that these books may be found in your university library.

Good luck!

The main difference is about the approach is used; when an implicit approach is used some simplifications  are made to reduce the computational process. I would suggest to read two useful documents provided by: 

1) Gernay T, Franssen JM. SAFIR MANUAL Materials SILCON_ETC and CALCON_ETC. University of Liege, Belgium: Environnement & Constructions Structural Engineering; 2011.

2) Franssen J-M. Plastic analysis of concrete structures subjected to fire. In: Pietro G. Gambarova, Roberto Felicetti, Alberto Meda, Riva P, editors. Fire Design of Concrete Structures: What now? What next? Milan, Italy: Milan University of Technology; 2005. p. 133-45.

Hope they help.

Hi Gayan,

Consider the simple heat equation.

Tt = Txx  ................................................................................…(1)                    

(with out mentioning the boundary conditions to keep it simple) 

Explicit Scheme:  Is one in which the differential equation is discretized in such a way that there is only one unknown on the LHS of the difference equation and it is computed in terms of all other terms on the RHS which are known. Let us write an Explicit Scheme for eqn (1):

Discretizing time derivative using forward difference and the second order derivative using central difference we get:

(Tjn+1 - Tjn )/ ∆t =  (Tj+1n – 2 Tjn + Tj-1n) / ∆x2

This can be simplified as:

Tjn+1 = (1-2r) Tjn + r(Tj-1n + Tj+1n) …………………………………………..(2)

 where r = ∆t/ ∆x2

This is an explicit scheme called FTCS (Forward differencing in Time and Central differencing in Space at time level n) for solving a 1-D heat equation. Explicit as we see one unknown on LHS (Tjn+1) being calculated in terms on all the term on RHS which are known as they are at previous time level n.

Explicit Schemes have to satisfy stability conditions in order for solution to converge. For example above Explicit Scheme in eqn (2) will be stable if  r ≤ ½.

Implicit Scheme:  Is one in which the differential equation is discretized in such a way that there are multiple unknowns at n+1 time level on the LHS of the equation and the terms on RHS are known ones at n time level. Let us write Implicit Scheme for eqn (1):

(Tjn+1 - Tjn )/ ∆t =  (Tj+1n+1 – 2 Tjn+1 + Tj-1n+1) / ∆x2                     

(discretizing 2nd order derivative at time level n+1 unlike Explicit Scheme where we discretized it at n level). We simplify it to:

(1+2r) Tjn+1 - rTj-1n+1 - rTj+1n+1 = Tjn………………………………………..(3)

Here we discretized time derivative term in backward differencing (Backward in Time) and discretized 2nd order term in Central differencing in Space both at time level n+1. Therefore this implicit scheme is called BTCS scheme.

Implicit Schemes result in multiple simultaneous algebraic equations (here we’ll have 3 equations corresponding to 3 unknowns at time n+1 level) that need to be solved as a system of equations of the form [K]{T}n+1 = {F}n. In Our simple case here we have [K]3x3 {T}3n+1 = {F}3n.

Good thing about Implicit Schemes is that they don’t have to satisfy any stability conditions. They are unconditionally stable. But they are computationally expensive as you need more storage space for [K] matrix and need sophisticated algorithms to solve.