We note i the space index in x direction, j the space index in y direction and n the time index. and are the numbers of point in each of these directions.
For that, one uses ADI Method (for Alternating Direction Implicit). Generally, this method is used to solve the compressible Navier-Stokes equations. When applied to the 2-D Burgers equation, this scheme becomes :
and and represent the second order central difference operator defined by :
This method is first-order accurate with a truncature error of and is unconditionally stable for the linear case. Obviously, a tridiagonal system of algebraic equations must be solved during each step. For that is used the gaussian elimination algoithm, sometimes refered to as the Thomas Algorithm (see Appendix A for more informations).
Principle of the ADI Method :
The algorithm initially consists in calculating on all the field the intermediate value starting from value calculated with the previous time step. In our case (arbitrary choice), we reverse the tridiagonal system with y fixed, and this for all including between 0 and 1 ().
Then, once traversed all the grid (all x and all y), one calculates (solution sought with the time step n+1) from calculated with the fictitious time step .
The method thus consists in systematically making a double sweeping with each time step. The algorithm of calculation is given in a diagrammatic way in Appendix B.