移流型方程式およびプリミティヴ方程式の数値解を求める際, 境界条件によって生ずる誤差

Abstract

The boundary errors in numerical integrations of hydrodynamical equations are discussed. If the advective type equation is solved on an open domain, adopting centered difference scheme, an extra boundary condition is needed at the outflow point and it brings some errors. It is shown that these errors result from the false reflection of a computational mode wave, synchronized with the incident physical wave. By assuming the situation that in a semi-infinite domain, the incident physical wave and the reflected suprious wave are in balanced state, the rate of reflection of the computational mode is estimated. It was found that if the quantity at the outflow boundary point is extrapolated from the interior of the domain with l-th order continuity, the reflection rate is tan (l+1)(p/2), +where p is the wave number of the incident wave measured in grid unit.The same method of analysis is applied to the examination of boundary conditions of primitive equations. It was revealed that adopting some boundary conditions, the reflection rates of computational mode of gravity wave exceed unity, while a certain condition does not bring artificial reflection at all. Discussions are made about the differences between the Platzman's analyses (1954) and the present results.