Derivation and dispersion analysis of compact schemes applied in differential equations with third derivative

Simulations and approximations of higher-order derivatives are essential to describe many complex phenomena in engineering and physics. To ensure stability and accuracy, compact numerical approximations are employed to solve many differential equations with problems involving dispersive waves. This work introduces a new class of implicit compact finite difference schemes that systematically generate fourth-, sixth-, and eighth-order accurate approximations for the third derivative. To detect many types of schemes with different orders of accuracy, the process of solving several systems of equations is required. The desired accuracy for each scheme is verified by employing special polynomials in the schemes and obtaining errors as the remaining terms for these polynomials. In addition to gaining the accuracy and stability of introduced schemes, achieving high resolutions makes them popular candidates when dealing with hyperbolic problems. From Fourier analysis, dispersive errors for the derived schemes are examined. These errors clearly illustrate the ability of high order compact schemes in capturing high resolutions. Finally, schemes proposed from this work are applicable to many computational problems and can be expanded to solve some types of multi-dimensional problems.