On the Error Estimates of a New Operator Splitting Scheme for the Navier-Stokes Equations with Coriolis Force
An operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations with Coriolis force. Under some mild regularity assumptions on the continuous solution, error estimates and the stability analysis for the velocity and the pressure of the new operator splitting scheme are obtained. Some numerical results are presented to verify the theoretical predictions.
In this paper, we consider the numerical approximation of the unsteady Navier-Stokes equations with Coriolis force: where is a bounded region in () with a sufficiently regular boundary . is the velocity field, is the pressure divided by the density (i.e., the kinematic pressure), is the kinematic viscosity coefficients, is the Reynolds number, is the vector of body forces, is the angular velocity vector, is the radius vector from the center of coordinates, and is the so-called Coriolis force.
For the sake of completeness, the equations should be supplemented with appropriate initial and boundary condition: The difficulties for the numerical simulation of the incompressible flows are mainly of two kinds: nonlinearity and incompressibility. The velocity and the pressure are coupled by the incompressibility constraint, which requires that the solution spaces, to which the velocity and the pressure belong, verify the so-called inf-sup condition. To overcome these difficulties, operator splitting method, which can be viewed as the fractional step method, is introduced. Fractional step methods allow to separate the effects of the different operators appearing in the equation by splitting the time advancement into a series of substeps. In addition, the cost of simulation can be also reduced by using the fractional step method. However, these methods have a main disadvantage that splitting error is inevitable unless the operator is commute.
The origin of this category of methods is contributed to the work of Chorin and Témam [1, 2]. They developed the so-called projection method, in which the second step consists of the projection of an intermediate velocity field onto the space of solenoidal vector field. The most attractive feature of projection methods is that, at each time step, one only needs to solve a sequence of decoupled elliptic equations for the velocity and the pressure, which makes it very efficient for large-scale numerical simulations. However, several issues related to these methods still deserve further analysis, and perhaps the most salient of these are the behavior of the computed pressure near boundaries and the stability of the pressure itself. The incompatibility of the projection boundary conditions may introduce a numerical boundary layer of size [3, 4], where is the kinematic viscosity and is the time step size. The end-of-step velocities of the projection do not converge in the space , since they do not satisfy the correct boundary conditions.
These methods have been widely investigated. Guermond et al. in  review theoretical and numerical convergence results available for projection methods. In [6–9], the analysis on first-order accurate schemes in the time size is presented. In [10, 11], Shen derived a second-order error estimates for the projection method. Olshanskii et al.  proposed a projection method for the Navier-Stokes equations with Coriolis force and study the accuracy of its semidiscrete form. In , a new discrete projection method for the numerical solution of the Navier-Stokes equations with Coriolis force is presented, where the scheme is treated as an incomplete LU factorization of the transition operator for fully implicit time discretization. In , complex 3D simulations of the Stirred Tank Reactor model by a modified discrete projection method for the rotating incompressible flow are presented. Numerical experiments from [13, 14] show that including -term in the second step enhances stability and accuracy of the scheme for the case of dominating Coriolis forces.
In this paper, we will consider the unsteady Navier-Stokes equations with Coriolis force (1.1). Using the technique developed in [7, 11] for the case of , a new scheme is introduced, which is a two-step scheme and allows to enforce the original boundary conditions of the problem in all substeps of the scheme. Some error estimates of both velocity and pressure for the proposed operator splitting scheme are given, which leads to the convergence of both the intermediate and the end-of-step velocities of the method to a continuous solution in the spaces and as in .
The remainder of this paper is organized as follows. In Section 2, we introduce some notations and assumptions, such as the regularity assumption for their solution. In Section 3, we present a new operator splitting scheme. In Section 4, the stability analysis is presented, then in Section 5, some error estimates for the intermediate, end-of-step velocity, and the pressure are given. Finally, in Section 6, some numerical results are presented to illustrate the theoretical results.
2. Function Setting
In order to study approximation scheme for problem (1.1), the following notations and assumptions are presented. we denote by and the inner product and norm in or . The spaces and are equipped with their usual norm; that is, The norm in will be denoted simply by . We will use to denote the duality between and for all .
The following subspace is also introduced: For the treatment of the convective term, the following trilinear form is given: It is well known that is continuous in , provided if , and this form is skew-symmetric with respect to its last two arguments, that is, In particular, we have and for , We also define the Stokes operator: where is an orthogonal projector in the Hilbert space onto its subspace . The Stokes operator is an unbounded positive self-adjoint closed operator in with domain , and its inverse is compact in . Having the following properties: there exists constant , such that , Furthermore, from (2.8), we will use as an equivalent norm of for .
For the purpose of this paper, we also need the following regularity assumptions: (A1) . In the three-dimension case, we assume additionally (A2) where (A2) is automatically satisfied with some appropriate constant when .
Under the regularity assumption (A1)-(A2), one can show that  (a) (b) .
In addition, if we also assume that (A3) holds, we have (c)
which will be used in the sequel. Indeed, the estimates (a-b) and the estimate (c) were proved for the Navier-Stokes without Coriolis term in [9, 16], respectively. However, adding linear skew-symmetric term to the momentum equation does not change arguments from , but leads to (a)–(c) with constant depending, in general, on . Next, we cite the following lemma, which will be frequently used.
Lemma 2.1 2.1 (Discrete Gronwall Lemma). Let , , , and be nonnegative sequences satisfying Assume and let then Hereafter, we will use to denote a generic constant which depends only on , and constants from various Sobolev inequalities. We will denote as a generic positive constant which may additionally depend on , , .
3. New Operator Splitting Scheme
Equation (1.1) can be written as such that So an algorithm can be formulated as follows: for and take as the approximate solution of (1.1) at time .
The scheme (3.3) has an irreducible splitting error of order . Hence, using a higher-order time stepping scheme does not improve the overall accuracy. So a first-order accurate semidiscrete version can be obtained as follows: let , we solve successively and by Note that we have omitted the dependency to of the function to simplify our notations; we will do so for .
As can be seen in (3.5), the main difference between this method and the standard projection method is the introduction of a viscous term in the incompressible step, which allows the imposition of the original boundary condition (2.6) on the end-of-step velocity . Similar ideas can be found in the -scheme  in which viscosity and incompressibility are also coupled, and some other methods such as [18–20], all of which involve an incompressible step with part of the viscous term. It leads to convergence of both the intermediate and the end-of-step velocities of the method to a continuous solution in space and . In comparison with the scheme, our scheme is two steps instead of three steps in scheme. Moreover, the fact that satisfies the correct boundary conditions will allow us to obtain improved error estimates comparative with the standard projection method.
Denoting the corresponding right-hand side by , at each time step, we have to solve the following two subproblems: with .
The first step of the method is a linearized elliptic problem, which can be seen as a linear Burger’s problem. The second step is a generalized Stokes problem. To solve problem (3.6), the fixed point iterative technique is used as in [21, 22], which is cheaper than the conjugate gradient method used by the least-square technique in the corresponding advective subproblems appearing in Glowinski’s -scheme. To solve problem (3.7), the efficient technique of the functional equation satisfied by the pressure is used, which is used in the Glowinski’s -scheme for the corresponding Stokes problem; that is, conjugate gradient method is applied on the variational formulation of such an equation. One defect of this method is that the discrete inf-sup compatibility condition should be satisfied.
4. Stability Analysis
For the sake of simplicity, we will only consider the homogeneous boundary condition , that is, for the scheme (3.4)-(3.5).
Theorem 4.1. Under the assumptions -, there exists a constant , such that
Proof. We take the inner product of (3.4) with to get Next, taking the inner product of (3.5) with , and using the condition , we obtain Combing (4.2) with (4.3), and using the Young’s inequality, we have Summing up the inequality (4.4) for , we obtain Applying the discrete Gronwall lemma to the above inequality, we obtain Thus, using the regularity properties of the continuous solution , for arbitrary , we have which means that From (4.2), it yields So we have . The proof is completed.
Remark 4.2. The formula (4.7) can be viewed as the discrete version of the classical energy estimate for the Navier-Stokes equations . From (4.6), we have This estimate provides a meaningful bound for for the first few time steps, that is, for small .
5. Error Analysis
In this section, we present an error analysis of the operator splitting scheme introduced in the previous section. Firstly, we define the semidiscrete velocity error as We give a first estimate for and which shows that and are both order approximations to in and .
Theorem 5.1. Under the regularity assumptions –, there exists a constant , such that
Proof. Let be the truncation error defined by
where is the integral residual of the Taylor series, that is,
By subtracting (3.4) from (5.3), we obtain
The nonlinear terms on the right-side can be split into three terms:
Taking the inner product of (5.5) with , using the identity () , we obtain
On the other hand, we derive from (3.5) that
Taking the inner product of the last equality with , and using , we obtain
Combing (5.7) with (5.9), we obtain
We bound each term in the right-hand side of (5.10) independently.
Taylor residual term: Pressure gradient term: Nonlinear term: The external term: The viscous term: The rotating term: Inserting the above estimates into (5.10), we obtain Summing up the inequality (5.17) for , we get By applying the discrete Gronwall lemma to the above inequality, we derive Using the regularity properties of , we obtain Finally, the bounds for follow from (5.20) and the triangle inequality. Theorem 5.1 is proved.
Remark 5.2. Theorem 5.1 shows that
since . Moreover, we also have
Next, we will use the previous result to improve the error estimates for the velocity and give an error estimate for the pressure as well. The result shows that both and are weakly first-order approximations to in .
Theorem 5.3. Under the regularity assumptions –, there exists a constant , such that
Proof. Taking the sum of (3.4) and (3.5), we obtain
Let us denote
Subtracting (5.25) from (5.3), we obtain
Taking the inner product of the expression (5.27) with , we obtain
together with (5.28), it yields
Similar to (5.6) for the nonlinear term, together with (5.30), it yields
We will focus on the right-hand side as follows.
The Coriolis term is estimated as follows: For the Taylor residual term, we have For the nonlinear term, it yields Using (2.6)-(2.8), we derive and together with Theorem 5.1, Similarly, Inserting the above inequality into (5.31), we obtain Taking the sum of the above inequality for , using the regularity assumption of the solution and Theorem 5.1, yields By applying the discrete Gronwall lemma to the last inequality, we obtain For , we have and together with (5.40), we derive (5.23).
Next, we derive the estimate for the pressure; we recast (5.27) as firstly, by using (2.6) and Theorem 5.3, for all , Using the Schwarz inequality, we have also, for all , Finally, we derive Therefore, by using Theorem 5.1 and (5.40), we obtain (5.24).
The error estimate of Theorem 5.3 can be improved to first order on the norms of and for the end-of-step velocities .
Theorem 5.4. Under the regularity assumptions –, for small enough , there exists a constant , such that
Proof. Taking the inner product of (5.27) with , we have
The estimates below are obtained on the right-hand term of (5.47):
For the nonlinear term, similar to (5.6), the below estimates are obtained:
For the remainder of nonlinear term, it yields
where we have used Theorem 5.1 and formula (2.6).
Taking the sum of the formula (5.47) for from 0 to , together with the above estimates, we get By virtue of the formula (5.23) and the regularity assumption (A2), (A3), we obtain For sufficiently small , we can take the last term to left side and apply the discrete Gronwall lemma to the last inequality, so the proof is completed.
Theorem 5.5. Under the regularity assumptions –, for small enough , there exists a positive constant , such that and , then
Proof. We shift the index in (3.5) to get and taking the sum with (5.5), we obtain Taking the inner product of (5.56) with , the left-hand term of (5.56) can be written as Now, we give the estimates of the right-hand term of (5.56): where we have used Theorem 5.1. Simultaneously,