An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burger’s type in two-dimensions. These nonlinear PDEs are essential because of the description of various mechanisms in engineering and physics. The nonlinear convective and advection are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization so that one discrete equation leads to the accuracy of orders three or four depending upon the choice of the grid network. The consistency and stability preserving third-order spatial accuracy and second-order accurate time discretization are described by Fourier analysis applied to the linearized error equations. The scheme is used for solving celebrated nonlinear PDEs, such as the non-degenerate convection-diffusion equation, generalized Burgers-Huxley equation, Buckley-Leverett equation, and Burgers-Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.

The Lie group of infinitesimal transformations technique and similarity reduction is performed for obtaining an exact invariant solution to generalized Kadomstev-Petviashvili-Boussinesq (gKPB) equation in (3+1)-dimensions. We obtain generators of infinitesimal transformations, which provide us a set of Lie algebras. In addition, we get geometric vector fields, a commutator table of Lie algebra, and a group of symmetries. It is observed that the analytic solution (closed-form solutions) to the nonlinear gKPB evolution equations can easily be treated employing the Lie symmetry technique. A detailed geometrical framework related to the nature of the solutions possessing traveling wave, bright and dark soliton, standing wave with multiple breathers, and one-dimensional kink, for the appropriate values of the parameters involved.