\begin{document}$ \theta $\end{document} scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations

In this article, a \begin{document}$ \theta $\end{document} scheme based on wavelet-like incremental unknowns (WIU) is presented for a class of porous medium diffusion-type equations. Through some important norm inequalities, we prove the stability of \begin{document}$ \theta $\end{document} scheme. Compared to the classical scheme, the stability conditions are improved. Numerical results show that the \begin{document}$ \theta $\end{document} scheme based on the WIU decomposition is efficient.


1.
Introduction. Incremental unknowns (IU) method which arises from the dynamical systems theory of Navier Stokes equations (see [9]) is a very powerful tool for the computation and analysis of fluid flows. The primary motivation of the IUs theory is to approximate inertial manifolds when finite differences are used for the spatial discretization (see [5]). However, Chen and Temam proved that IU allows to define hierarchical preconditioners for discrete operators arising from elliptic-like problems (see [1], [2]). Many studies on IU have been carried out (see [8], [10]- [15]). Wavelet-like incremental unknowns (WIU) deserve special attention because they enjoy the L 2 orthogonality property between different levels of unknowns. This makes the method with multilevel wavelet-like incremental unknowns particularly appropriate for the approximation of evolution equation. Chen and Temam applied the multilevel WIU to a Reaction-Diffusion equation (see [3]), they presented several numerical schemes using one dimensional WIU. The fully discretized explicit and semi-explicit schemes for the reaction-diffusion equation are presented and analyzed, the stability conditions are improved with the corresponding algorithms. The author in [13] established two semi-implicit schemes with multilevel WIU methods for the same equation, the stability conditions of the two schemes become better. [8] proposed a new type of WIU method for the two-dimensional reaction-diffusion equations with a polynomial nonlinear term and analysed the stability of Euler explicit and semi-implicit schemes.
The purpose of this paper is to apply WIU to two dimensional reaction-diffusion equations with more general nonlinear term and present a θ Scheme which includes explicit scheme, implicit scheme and Grank-Nicolson scheme. We will prove some important inequalities, then we prove that under suitable conditions, the θ scheme is stable. Compared with the classical theme discretized without WIU, the time step can be larger. Numerical result shows that the θ scheme save more CPU time than the scheme without WIU.
The paper is organized as follows. In Section 2 we present the prorous medium diffusion-type equation and its finite difference discretization. Then in Section 3 we recall the definition of the two dimensional WIU and the multilevel discretization in space. New θ schemes based on multilevel WIU are established in Section 4. In Section 5, firstly we write the variational form of the approximate scheme, then we develop the stability criteria of the schemes. Finally, numerical results show the efficiency of the new methods.

Dynamic Equation and Discretization.
Let Ω be an open-bounded subset of R n . We consider the model P : under the following assumptions: (H1) For each ξ ∈ R, the map (t, x) → g(t, x, ξ) is measurable and almost everywhere in Ω × R ξ → g(t, x, ξ) is continuous and differentiable.
(H2) We assume that there exist c 1 > 0, c 2 > 0 and c 3 > 0 such that where a : R + → R + is an increasing function and the sign function is defined as follows: (H3) There exists c 7 > 0 such that for almost every (t, x) ∈ R + × Ω : ξ → g(t, x, ξ) + c 7 ξ is an increasing function.
(H4) u 0 ∈ L 2 (Ω). Using the above assumptions, we have the following existence and uniqueness result(see [4]): Theorem 2.1. Under the assumptions (H1) to (H4), there exists a unique solution u ∈ L q (0, T ; L q (Ω)) L 2 (0, T ; H 1 0 (Ω) L ∞ (η, T ; L ∞ (Ω)), ∀ η > 0, satisfies P. Now, we consider spatial discretization by finite difference with mesh size h d = 1/(2 d N + 1), where N ∈ N. For the sake of simplicity, here let Ω = [0, 1] 2 . We have where U h d ∈ R (2 d N ) 2 is the vector that consists of approximate values of u at the grid points and X = (x 1 , x 2 , ..., FOR POROUS MEDIUM DIFFUSION-TYPE EQUATIONS   463 has the form Here, I is the identity matrix, C and I are both of order 2 d N .
For simplicity, we write 3. Multilevel Wavelet-like Incremental Unknowns. We introduce the waveletlike incremental unknowns equation (3) by recalling the definition of WIU in [3]. We separate evenly the unknowns into four different parts according to the grid, see Fig.1. Here u d 2i,2j is the value of u corresponding to coarse grid M 1 , u d 2i−1,2j , u d 2i,2j−1 and u d 2i−1,2j−1 are the values of u corresponding to complementary grids A 1 , A 2 and A 3 .
Inversely, we have

YANG WANG AND YIFU FENG
We reorder U d intoŨ d by letting where are column vectors ordered in lexical order (from left to right, from down to up) separately by four different types of grid points Here P d is a permutation matrix of order (2 d N ) 2 which changesŨ d to U d and P d has the following form .
Here 0 denote the zero matrix of the order 2 d N × 2 d−1 N . S d is the transfer matrix of order (2 d N ) 2 which transfersŪ d toŨ d , S d has the following form where the I d−1 is the identity matrix of order (2 d−1 N ) 2 . Substituting (8) into the finite difference equation (3) and multiplying the equation Noting that P T d P d = I d , P T d and g can commute, we obtain θ SCHEME FOR POROUS MEDIUM DIFFUSION-TYPE EQUATIONS 465 which is the 2-level WIU scheme. The next level of WIU on Y d can be introduced by repeating the same procedure. We now separate Y d into four parts and denote thatȲ Therefore, we can obtain the equality with and Here, I d−2 denote the identity matrix of order (2 d−2 N ) 2 . P d−1 , S d−1 have the similar structures as P d , S d respectively but they are both matrices of order Noting thatP d−1 ,S d−1 are matrices of order (2 d N ) 2 , we can find Substituting (13) into (10) and multiplying the equation by (P d−1Sd−1 ) T , we can see that Here, S = P d S dPd−1Sd−1 . Generally, for l = d − 1, d − 2, · · · , 1, we can introduce the next level of WIU on Y l+1 by repeating the same method. Let We can see that then we can writeŪ · · · , 1 and using the same method, we can obtain the d + 1-level WIUs in terms of Y and Z, Here 4. Approximate Scheme. We now propose θ scheme based on WIU introduced in section 3. Firstly, we find a method to compute the nonlinear term According to some computations, we get using Tylor expansion at Y d and neglecting the terms For two level wave-like incremental unknowns, the form of the equation (19) with Using the same approximation as (20), the approximate equation above becomes In fact, Using the above approximation (20) and the definition ofS d−1 ,P d−1 , we have Finally, the (d + 1)-level incremental unknowns equation (19) is approximated by Now as for time discretization, we propose θ scheme as follows Here, A * = S T A d S, τ is the time step and θ ∈ [0, 1] is a parameter.
5. Stability Analysis of θ scheme. In this section, firstly, we write the equivalent variational formulation of scheme (25), then under suitable conditions, we present the stability analysis of θ scheme (26). Let V h d (or simply V d ) be the function space spanned by the basis functions w h d ,Mij (x), Here M ij = (ih d , jh d ), i = 1, 2, · · · , 2 d N, j = 1, 2, · · · , 2 d N , and w h d ,Mij (x) satisfies We introduce two finite difference operators ∇ 1,h d and ∇ 2,h d : where e 1 = (1, 0), e 2 = (0, 1). We define the following discrete scalar product where (·, ·) is the scalar product in L 2 (Ω). Let · h d = ((·, ·)) 1 2 h d and observe that · h d and | · | are Hilbert norms on V d . With the help of step functions, we can write the finite difference discretization scheme (3) in a variational form Scheme (3) can be recovered by choosingũ = w h d ,Mij (x). We now separate space V d into two spaces Y d and Z d according to the definition of wavelet-like incremental unknowns. Let Y d as the space spanned by the basis functions Ψ 2h d ,M2i,2j (x), i = 1, 2, · · · , 2 d−1 N, j = 1, 2, · · · , 2 d−1 N , and Let Z d as the space spanned by the basis function χ h d ,A (x), there are three kinds of points of A. For i, j = 1, 2, · · · , 2 d−1 N , 0, otherwise.
From the definition of Ψ 2h d ,M (x) and χ h d ,Ai (x), for ∀ z d ∈ Z d , ∀ y d ∈ Y d , we can obtain three conclusions: (1) The decomposition of V d makes (5) hold.
Thus the space V d can be decomposed as Obviously, for ∀ u d ∈ V d , we have With above decomposition, we can see that the variational form is identical to (22). Multilevel incremental unknowns can be recovered in a similar fashion, we decompose Y l , l = d, ..., 1 into Remember that Y d+1 ∈ Y d+1 = U d . Therefore for any function u d ∈ V d , we can write it as u d = y + z, where y = y 1 ∈ Y = Y 1 and z ∈ Z = Z 1 ⊕ Z 2 ⊕ · · · ⊕ Z d . Using the above decomposition, we can prove that the following variational form is identical to (25).
Before presenting the stability theory, let us introduce some useful lemmas.

YANG WANG AND YIFU FENG
Proof. Firstly, we prove the right hand side of (33). Due to the Dirichlet boundary condition, we agree u αβ = 0 if α or β = 0 or 2N . For every function u h ∈ V h , we have It is easy to see that Secondly, we prove the left hand side of (33). Thanks to Cauchy's inequality, we have Then we obtain Using the definition of | · | and (34), we have Thus, the inequality (33) holds. jh 1 ), ∀ y ∈ Y is a step function, we have the following equality Here, Ω denotes any small square in K 1 ij . The proof is completed.
Lemma 5.3. For every function y ∈ Y, Here |y| ∞ is the maximum norm of y. jh 1 ), Using the expression of y, we have Then we prove (37). Using the definition of . h d , |.|, Lemma 5.2 and (a − b) 2 ≤ 2(a 2 + b 2 ), it suffices to prove that 8 y 2 (ih 1 , jh 1 ) and By simple computations, we prove (37) holds.
Lemma 5.4. For given time step τ = T K , if cτ < 1, then for ∀n ≤ K, we have Now, we have the following theorem for θ scheme.
• Assuming the conclusion is correct up to k = n.