A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains. Trivial conforming and non-conforming Cartesian meshes with 3-point stencils in the 1-D case and 9-point stencils in the 2-D case are used in calculations. The numerical solutions of the 1-D wave equation as well as the 2-D wave and heat equations for a simple rectangular plate show that the accuracy of the new approach on non-conforming meshes is practically the same as that on conforming meshes for both the Dirichlet and Neumann boundary conditions. Moreover, very small distances (\(0.1 h - 10^{-9}h\) where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not decrease the accuracy of the new technique. The application of the new approach to the 2-D problems on an irregular domain shows that the order of accuracy is close to four for the wave and heat equations and is close to five for the Poisson equation. This is in agreement with the theoretical results of Part 1 of the paper. The comparison of the numerical results obtained by the new approach and by FEM shows that at similar 9-point stencils, the accuracy of the new approach on irregular domains is two orders higher for the wave and heat equations and three orders higher for the Poisson equation than that for the linear finite elements. Moreover, the new approach yields even much more accurate results than the quadratic and cubic finite elements with much wider stencils. An example of a problem with a complex irregular domain that requires a prohibitively large computation time with the finite elements but can be easily solved with the new approach is presented.

在第1部分中开发了一种新的数值方法，用于求解与时间相关的波动和热方程以及与时间无关的Poisson方程，用于在规则和不规则域上模拟一维和二维测试问题。在计算中使用在1D情况下具有3点模板和在2D情况下具有9点模板的琐碎的符合和不符合要求的笛卡尔网格。一个简单矩形板的一维波动方程以及二维波动方程和热方程的数值解表明，新方法在不合格网格上的精度实际上与在合格网格上的精度相同Dirichlet和Neumann边界条件。此外，非常小的距离（\（为0.1h - 10 ^ { - 9}ħ\）其中ħ笛卡尔网格的网格点与边界之间的网格大小）不会降低新技术的准确性。新方法在不规则域上的二维问题中的应用表明，波动和热方程的精确度阶次接近于4，而泊松方程的精确度阶次接近于5。这与本文第1部分的理论结果一致。通过新方法和有限元方法获得的数值结果的比较表明，在相似的9点模板上，新方法在不规则区域上的波动和热方程的精度高2阶，泊松方程的精度高3阶。比线性有限元 此外，与具有较宽模板的二次和三次有限元相比，新方法产生的结果要准确得多。给出了一个复杂的不规则域问题的示例，该问题需要有限的计算时间，但是使用新的方法可以轻松解决。