The Kadomtsev-Petviashvili (KP) equation is a two-dimensional dispersive wave equation which was proposed to study the stability of one soliton solution of the KdV equation under the influence of weak transversal perturbations. It is well know that some closed-form solutions can be obtained by  function which have a Wronskian determinant form. It is of interest to study KP with an arbitrary initial condition and see whether the solution converges to any closed-form solution asymptotically. To reveal the answer to this question both numerically and theoretically, we consider different types of initial conditions, including one-line soliton, V-shape wave and cross-shape wave, and investigate the behavior of solutions asymptotically. We provides a detail description of classification on the results. The challenge of numerical approach comes from the unbounded domain and unvanished solutions in the infinity. In order to do numerical computation on the finite domain, boundary conditions need to be imposed carefully. Due to the non-periodic boundary conditions, the standard spectral method with Fourier methods involving trigonometric polynomials cannot be used. We proposed a new spectral method with a window technique which will make the boundary condition periodic and allow the usage of the classical approach. We demonstrate the robustness and efficiency of our methods through numerous simulations.