Finite Volume Method For Two Phase Flow Problems

The finite Volume Method, also called as Generalized Difference Method, was firstly put up by professor Li Yonghua in 1982.The two phase problem is that pressure and saturate are suitable to the equation which is besides oil and water in the liquid. The concrete equation is:First, we divide the area into a series of triangles, and adopt circumcenter dual decomposition. Then the trial function space is chosen as the linear element space. The test space is chosen as the piecewise constant function space.We adopt the trial function space U h and the test function space V h above, then the scheme of the finite volume method is: find P_h∈U_h, that:Or equally, Where According the typist of P_h is linear piecewise:This is the partial difference equation of P₀ .Another thing is the We may reach the approximate equation isWe combine the (0-3)and (0-4), therefore we get the approximate value of the pressure at the node.The solution to saturate equation is following:We will use two methods to solve this equation. We divide ?Ωinto two parts:Where n stands for the unit outer normal vector of ?Ω. In my essay we adopt the first order space, due to the discontinuity of V h on the boundaries of the dual elements, one can not apply the Galerkin finite element method on the entire regionΩ. But it is feasible to apply it on a single dual element K_P0^*. So we seek S_h (? , t)∈V_h satisfyingUse the integral formula, we could reachSimilarly as in (0-6) we can defineThey are referred to as the upwind and the downwind values of S h at x∈?K_P0^*, respectively. On the analogy of the classical upwind scheme, we replace S_h in the line integral of the left-hand side of (0-8) by S_h⁺ to obtainThen we obtain a semi-discrete upwind scheme: In actually computation, we choose 1, x - x₀,y-y₀respectively on every dual partition. That means we will have three equations on every point. Various kinds of finite difference quotients can used to further discretize the time th, such as forward difference, backward difference, or Crank-Nicolson difference.Now we narrate the second method to solve the saturate equation. We will construct the finite element space. We also adopt the pressure partition. In the second method, the trial function is continues globally. It is determined by the three points on every piece.So, we multiply v h on both side of the saturate equation, and integral on K_P0^*:From (0-11), we may know, we put every basis functions v_hinto (0-11), and we will reach:We also use upwind scheme:Therefore, on every node it will give a linear equation. From all the equations, we will get the approximate value of the saturate on every node. Otherwise, th need to discrete, we may use forward difference, backward difference and Crank-Nicolson difference. One point should be noticed that we should use numerical integral formula of high accuracy in computing . At last, both the arithmetic are convergent through numerical examples.