As the finite difference methods or other methods of discreet variables both in time andspace coordinates are used to solve the initial value problems. One of the sufficient conditionsfor computational stability is to maintain the conservation or boundness of the total energy ofthe original physical problems in the computational schemes. Those schemes, which do not exactlyguarantee the conservation or boundness of energy, have always some special computational un-stable solutions. Until recently, all published and the so-called energy-conservative finite dif-ference schemes for integrating the dynamical equations of compressible fluid or the meteoro-logical primitive equations conserve the total energy only instantaneously, i.e. when the differen-tial in time remained, but cannot exactly conserve it as the finiie difference in time is also usedinstead of the differential. In this paper, by using some transformations of independent (co-ordinates) or dependent variables, we have successfully designed some time-space finite differ-ence schemes guaranteeing the energy and mass conservations perfectly. Thus the problem ofcomputational stability is solved completely. In order to solve these finite difference equations, some split methods have been developedalso. They possess the internal consistency and reduce the original problem into successivelysolving several onedimensional initial value problems, to which the economical method of fac-torization can be applied effectively. Thus, both computational stability and economy are guar-anteed. These methods can be used in routine. As the finite difference methods or other methods of discreet variables both in time and space coordinates are used to solve the initial value problems. One of the sufficient conditions for computational stability is to maintain the conservation or boundness of the total energy of the original physical problems in the computational schemes. which do not exactly guarantee the conservation or boundness of energy, have always some special computational un-stable solutions. Until recently, all published and so-called energy-conservative finite dif-ference schemes for integrating the dynamical equations of compressible fluid or the meteoro-logical primitive equations conserve the total energy only instantaneously, ie when the differen-tial in time remained, but can not exactly conserve it as the finiie difference in time is also usedinstead of the differential. In this paper, by using some transformations of independent (co-ordinates) or dependent variables, we have successf ully designed some time-space finite differ-ence schemes guaranteeing the energy and mass conservations perfectly. Thus the problem ofcomputational stability is solved completely. In order to solve these finite difference equations, some split methods have been developedalso. They possess the internal consistency and reduce the original problem into successivelysolving several onedimensional initial value problems, to which the economical method of fac-torization can be applied effectively. Thus, both computational stability and economy are guar-anteed. These methods can be used in routine.