Highly accurate compact mixed methods for two point boundary value problems

Finite difference method, and finite element method are widely used partly for their simplicity, though these methods can obtain first-order or second-order of accuracy. In this paper, we give two highly accurate (fourth-order and sixth-order of accuracy respectively), while still quite simple schemes for two point boundary value problems. We call them compact mixed methods, since they can obtain numerical solutions for unknown function and its first derivative simultaneously. To yield best numerical solutions, we combine our compact mixed schemes with a direct solver and an iterative solver. Then we employ Fourier method to analyze differencing errors shows that our compact mixed formulae are closer to the true wavenumber. Numerical experiments show compact mixed schemes are very efficient, and highly accurate techniques.