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In this paper, a new ordinary differential equation numerical integration method is successfully applied to various mathematical branches such as partial differential equation (PDE) boundary problems, PDE initial-boundary problems, tough nonlinear equations, and so forth. The new method does not use Jacobian, so it can handle very large systems, say the dimension N=1 000 000, or even larger. In addition, we give a very simple accelerating convergence approach for the linear algebraic equations arising from linear PDE boundary problems. All the numerical results show that the new method is very promising for super large scale systems.