Hidrológiai Közlöny 1988 (68. évfolyam)

6. szám - Gáspár Csaba: A konvektív diffúziós operátor matematikai vizsgálata

GASPAR CS.: Konvektív diffúziós operátor 347 convective and diffusion operators are played by appropriate bilinear functionals (Eqs. 28—33). The operator splitting method is also examined for both steady and unsteady problems. Using the contraction principle, it is shown that for sufficiently small time steps the operator splitting method is convergent (for steady problems) and stable (for unsteady problems). See Eqs. 19—25. Moreover, a modification of this method makes it unconditionally stable (Eq. 42). In special cases, using certain function transforms one can deduce the original convective diffusion problem to a simpler form which might be treated more easily from numerical point of view, as well. Three cases are investigated. If the velocity field of the convection is irrotational, i. e. it has a velocity potential (Eq. 34), then a simple multiplication by an exponential factor leads to a Helmholz equation (Eq. 35) which does not contain the first derivatives of the unknown function. As it is pointed out, this exponential factor may cause great numerical errors if the transport is not diffusion-dominant. Two other transformations is given for the one-dimensional case. All of the above transformations lead the original differential operator back to a self-adjoint one, which may be useful in the case of unsteady problems where the exact solution can be written in Fourier series with respect to the eigenfunctions of the operator (Eqs. 3(i

—40), provided that the determination of these eigenfunctions requires only an acceptable computational cost. Keywords: partial differential operator, diffusion problem, existence, uniqueness, operator splitting GÁSPÁR CSABA szakmai munkásságának összefoglalóját a Hídról. Közi. 1988/4. számában közöltük.