A new two-level implicit discretization of for the solution of singularly perturbed two-space dimensional non-linear parabolic equations

We propose a new two-level implicit difference method of O ( k 2 + kh 2 + h 4 ) for the solution of singularly perturbed non-linear parabolic differential equation ε ( u xx + u yy ) = f ( x , y , t , u , u x , u y , u t ) , 0 < x , y < 1 , t > 0 subject to appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are grid sizes in time and space directions, respectively, and ε > 0 is a small parameter. We also develop new methods of O ( kh 2 + h 4 ) for the estimates of ( ∂ u / ∂ x ) and ( ∂ u / ∂ y ) . In all cases, we use 9-spatial grid points and a single computational cell. The proposed methods are directly applicable to singular problems. We do not require any special scheme to solve singular problems. We also discuss alternating direction implicit (ADI) method for solving diffusion equation in polar cylindrical coordinates. This method permits multiple use of the one-dimensional tri-diagonal algorithm with a considerable saving in computing time, and produces a very efficient solver. It is shown that the ADI method is unconditionally stable. Numerical experiments are conducted to test the high accuracy of the proposed methods and compared with the exact solutions.