This paper describes a novel implicit method of order 2 in y- and 3 in x-direction in exponential form, by exploiting off-step discretization to solve numerically $2D$ non-linear elliptic partial differential equations in a rectangular region. We use variable mesh in x-direction and constant mesh in y-direction in order to solve convection–diffusion equation for large values of the coefficient of convection term. This method uses 9-point compact stencil. Detailed derivation and convergence procedure of the proposed method have been discussed. The method has been generalized to solve non-linear elliptic equations in vector form. The method is validated on several benchmark problems showing that formulation produces satisfactory results.

本文描述了一种新的隐式隐式方法，即y方向2 阶和x方向3阶指数形式，通过利用离步离散化进行数值求解$2D$矩形区域中的非线性椭圆偏微分方程。我们在x方向使用可变网格，在y方向使用恒定网格，以便求解对流项系数较大的对流扩散方程。此方法使用 9 点紧凑型模板。已经讨论了所提出方法的详细推导和收敛过程。该方法已推广到求解向量形式的非线性椭圆方程。该方法在几个基准问题上得到验证，表明配方产生了令人满意的结果。