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The existing 2-D alpha-mu scheme and alpha-epsilon scheme based on the method of space-time conservation element and solution element, which were constructed for solving the linear 2-D unsteady advection-diffusion equation and unsteady advection equation, respectively, are tested. Also, the alpha-epsilon scheme is modified to become the V-E scheme for solving the nonlinear 2-D inviscid Burgers equation. Numerical solutions of six test problems are presented in comparison with their exact solutions or numerical solutions obtained by traditional finite-difference or finite-element methods. It is demonstrated that the 2-D alpha-mu, alpha-epsilon, and nu-epsilon schemes can be used to obtain numerical results which are more accurate than those based on some of the traditional methods but without using any artificial tuning in the computation. Similar to the previous 1-D test problems, the high accuracy and simplicity features of the space-time conservation element and solution element method have been revealed again in the present 2-D test results.