Combined Compact Difference Research Articles

A fast solver of the shallow water equations on a sphere using a combined compact difference scheme

The spectral-like three-point combined compact difference (sp-CCD) scheme is developed for the shallow water models on a sphere, which is originally proposed by Chu and Fan (J. Comput. Phys. 140 (1998) 370). A standard test set proposed by Williamson et al. (J. Comput. Phys. 102 (1992) 211) is used to evaluate the accuracy of the new method. It is shown that the accuracy of the method is comparable to that of the spherical harmonics model as a whole. The relative efficiency of the method with respect to the spherical harmonics model appears from the resolution of 512×256, and it becomes significant for the higher resolutions. The test results suggest that the method could be extended to numerical atmospheric models.

A three-point sixth-order staggered combined compact difference scheme

A three-point, sixth-order, staggered, combined compact difference (SCCD) scheme is proposed for numerical models. The SCCD scheme is a generalization of the previously proposed combined compact difference (CCD) scheme, and has major improved features such as error and computational (CPU) time reduction especially for odd-order difference equations with odd-number boundary conditions. For nonperiodic boundaries, an additional sixth-or fifth-order boundary condition is proposed. The stability of the SCCD scheme is studied using the eigenvalue analysis.

A Three-Point Sixth-Order Nonuniform Combined Compact Difference Scheme

A three-point nonuniform combined compact difference (NCCD) scheme with sixth-order accuracy is proposed for numerical models. The NCCD scheme is a generalization of the previously proposed combined compact difference (CCD) scheme with a global Hermitan polynomial spline and has major improved features such as error and computational (CPU) time reduction. For nonperiodic boundaries, additional sixth- or fifth-order nonuniform boundary conditions are proposed. The NCCD scheme with either sixth- or fifth-order additional boundary conditions can increase the accuracy and decrease the CPU time about 1–2 orders of magnitude, compared to the CCD scheme.

A Three-Point Combined Compact Difference Scheme

A new three-point combined compact difference (CCD) scheme is developed for numerical models. The major features of the CCD scheme are: three point, implicit, sixth-order accuracy, and inclusion of boundary values. Due to its combination of the first and second derivatives, the CCD scheme becomes more compact and more accurate than normal compact difference schemes. The efficient twin-tridiagonal (for calculating derivatives) and triple-tridiagonal (for solving partial difference equation with the CCD scheme) methods are also presented. Besides, the CCD scheme has sixth-order accuracy at periodic boundaries and fifth-order accuracy at nonperiodic boundaries. The possibility of extending to a three-point eighth-order scheme is also included.