Parabolic Problems Discontinuous Finite Volume Element Method Numerical Simulation

In the frst chapter, we propose an upwind discontinuous fnite volume elementmethod for the following conservative convection-difusion equation, By combing upwind technique and discontinuous fnite volume element method pro-posed by Xiu Ye. This method inherits the advantages of upwind method anddiscontinuous fnite volume method, such as the high-precision, high-parallel, localconservation and the simple structure of space, etc. Error estimates in L2-norm and|||·|||1,h-norm for this scheme are derived.Then, in the second chapter, we propose an discontinuous fnite volume elemen-t method for the two-dimensional unsaturated soil water movement mathematicalmodel[45]:where Q(x, z, t) is cubic water-bearing rate, Sris root’s bibulous rate, K(Q) ishydro-power conduction coefcient, D(Q) is soil water difusivity, hydraulic con-ductivity coefcient K and water difusivity D with Q are related as follow s: where Qsand Qrare respectively saturated moisture and residual moisture of thesoil moisture, and0<Qs<1, saturated water conductivity Ks, soil parameters band the saturated soil water potential ψshave relation with soil structure, they areall known constants. So, K(Q), K(Q) Q)Here are models of fxed boundary conditions:(1)Initial conditions Q(x, z,0)=Q0, where Q0is the initial moisture content.(2)Boundary conditio nswhere Q0is the initial moisture content, Qsis the saturated moisture content.For the convenience of the study, it is assumed that area is a bounded domainin R2a proper and smooth enough, and in the boundary of, Q0andQsare all ze-ros. For this equation presents a numerical simulation method: discontinuous fnitevolume element method. The method does not require the function to maintain acontinuous requirement across the interior of the unit boundary, making space fora simple structure, and also has the advantages of high accuracy, high parallel, thisis an efective treatment method for such problems. In this section, we give theproblem discontinuous fnite volume element, we obtain the optimal estimates ofL2-norm and|||·|||1,h-norm. Finally, we give a parabolic equation numerical examplesto verify the efectiveness of this method.