Diagonal inversion of lower-upper implicit schemes

Abstract

A new diagonally inverted lower-upper (LU) implicit scheme is developed for the three-dimensional Euler equations. The matrix systems that are to be inverted in the scheme are treated by local diagonalizing transformations that decouple them into systems of scalar equations. Unlike the diagonalized alternating direction implicit method, the time accuracy of the LU scheme is not reduced since the diagonalizing procedure does not destroy time conservation. Furthermore, this diagonalization reduces the computational effort required to solve the implicit approximation and therefore transforms it into a more efficient method of numerically solving the three-dimensional Euler equations. I NCREASINGLY, attention is being directed towards implicit schemes as the need to develop more efficient numerical methods becomes apparent. An implicit approximation is an attractive way to increase the efficiency of the time-marching technique because implicit schemes permit much larger time steps than most explicit methods normally allow. The drawback of most implicit schemes is that they are computationally more expensive, per time step, than explicit methods. As a result, interest is being focused upon decreasing the computational effort required to solve the implicit approximation. Examples of schemes that address this issue are Chaussee and Pulliam's diagonalized alternating direction implicit (ADI) scheme1 and Obayashi and Fujii's lower-upper (LU) alternating direction implicit scheme.2 The Euler equations can be discretized into an implicit approximation that, when written in delta form, produces a large block-banded matrix system that is impractical to solve without first being approximately factored. In the present work, the LU implicit multigrid algorithm developed by Yokota and Caughey3 for the three-dimensional Euler equations has been made more efficient through a local diagonalizing procedure that reduces the computational effort required to solve the LU approximation . The LU factorization produces two block triangular operators (one upper and one lower) which, through back substitution, can be solved by effectively explicit sweeps that require matrix inversions at every mesh cell in the domain. The efficiency of this LU scheme can be increased by reducing the computational costs associated with these matrix inversions. This reduction can be achieved by a local diagonalizing transformation that decouples the matrix systems into scalar equations. Time conservation and stability remain unaltered by the decoupling since the LU scheme's differencing operators are unaffected by the diagonalizing transformations. This result is unlike the diagonalized ADI method, which is produced at the expense of time accuracy. The Euler equations for the generalized coordinate system ,?7,f) can be written