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The Backward Euler Fully Discrete Finite Volume Method for the Problem of Purely Longitudinal Motion of a Homogeneous Bar
We present a linear backward Euler fully discrete finite volume method for the initial-boundary-value problem of purely longitudinal motion of a homogeneous bar and an give optimal order error estimates in and norms. Furthermore, we obtain the superconvergence error estimate of the generalized projection of the solution in norm. Numerical experiment illustrates the convergence and stability of this scheme.
We consider the following mixed boundary-initial value problem: This problem (1.1) arises when one considers the purely longitudinal motion of a homogeneous bar . The displacement of the cross-section of the bar at time is denoted by . When both ends of the bar are fixed, , , that is the boundary condition. and are the initial data.
In theoretical analysis, the problem (1.1) was first treated by Greenberg et al. , by assuming that the function was monotonic, that is, and that the initial data was smooth, specifically Under these assumptions they showed the existence of a unique smooth solution which decays the zero solution as . (See also Greenberg  and Greenberg and MacCamy .) Andrews  made the hypotheses that , , the function is locally Lipschitz continuous, and there exists a constant such that whenever ; then he proved the existence of a unique global weak solution. Under the hypotheses that , and there exists a constant such that , , Y. Liu and D. Liu  proved the existence of a unique global strong solution of the problem (1.1).
Numerical simulation methods for the problem (1.1) are recently studied by several authors ([6–11]). In , Gao et al. studied a finite difference method of the problem (1.1) in the domain and proved the convergence of the method by using discrete functional analysis and prior estimate. In [7, 8], Jiang et al. studied two finite element methods of (1.1) and obtained the optimal error estimates of this finite element scheme in and norms. In , Z. Jiang and Y. Jiang, and in , Jiang and Li, studied a mixed finite element method and a expanded mixed finite element method, respectively, and obtained the optimal error estimates of these schemes. However, few work on finite volume methods of (1.1) was found (see ). As we know, finite volume methods (also called generalized difference methods) were proposed in eighties last century and developed very quickly. Now this kind of method becomes one of the main numerical methods for solving differential equations, for example, convection diffusion equations [12–14] and Navier-Stokes equations .
In this paper we want to make further study of finite volume methods for the problem (1.1). First in Section 2, we derive a finite volume weak form of (1.1) in the case that is nonlinear, then propose a linear backward Euler fully discrete finite volume scheme of (1.1). Existence and uniqueness of the solution of this scheme are proved. Next, in Section 3, we give optimal error estimates in and norms and superconvergence in norm by using new defined projections. Numerical experiments and computational results are presented in Section 4, which confirm our theoretical analysis.
2. The Linear Backward Euler Fully Discrete Finite Volume Scheme
In this section, we construct the finite volume method of the problem (1.1) and prove the existence and uniqueness of the solution of this finite volume scheme.
Firstly, let be a partition for the interval , with its nodes . The length of the element is denoted by , , is maximum of . We suppose is regular, that is, there exists a positive constant such that , . For the definition of the finite volume scheme, the dual partition of is needed, which is . The dual elements are denoted by , , , , where , .
Secondly, we define the piecewise linear trial function space over the partition , satisfying , where is the Sobolev space on . Then for all , where is the basis function associated with the nodes , It is easy to know that the derivative of with respect to is The test function space associated with the dual partition is defined as the set of all piecewise constants with for all . We may choose the basis function of in such a way that is the characteristic function of , that is, Then for any can be expressed as .
Obviously, Meanwhile, .
Thirdly, for the time interval , we give an isometric partition and denote the nodes , , .
We introduce some notations for functions and : Then we can get
Let be the solution of (1.1). Integrating (1.1) a over the dual element , we obtain where , , , and . The problem (2.7) can be rewritten in a variational form. For any arbitrary , we multiply the integral relation in (2.7) by and sum over all to obtain where for any arbitrary , the bilinear forms , are defined by
Since is a nonlinear function, we will consider the following linear finite volume scheme: find , such that where and are the interpolation projection of and onto the trial function space , respectively, and the interpolation operator is defined as We also need to introduce the interpolation , defined by, By Sobolev’s interpolation theory, we know that In this paper we adopt the standard notation for Sobolev space on with norm and seminorm . In order to simplify the notations, we denote by and skip the index when possible, that is, , , . We denote by the Banach space of all integrable functions from into with the norm for and the standard modification for . After all these denotations, we give the existence and uniqueness of the solution of the finite volume scheme (2.10).
Theorem 2.1. The solution of the finite volume scheme (2.10) is existent and unique.
For , choosing , . By (2.2) we have Using (2.10) we obtain that is, Noticing , the coefficient matrix of the system (2.16) is strictly diagonally dominant matrix. So, when and are known, the solution is existent and unique. Combining the above conditions, the solution is existent and unique when and are known. This completes the proof.
3. Error Estimates
In this section, we will prove the optimal error estimates in the and norms as well as the superconvergence error estimates in the norm. This needs some assumptions about the data. The nonlinear function satisfies The initial functions and satisfy The solution of (1.1) satisfies
For any , , , where , , . Using the definition of the interpolation operator we know that , . Noting and [11, Lemma 3.1] we have Hence So, when , we can get . Let . By [11, Lemma 3.2], we have On the other hand, using [11, Lemma 3.3], we also have For error estimate, we will use the generalized adjoint finite volume element projection of the solution of (1.1), that is, satisfies Let and denote the solutions of (1.1) and (3.8), respectively. Under the assumptions , , and together with [11, Theorems ] and the one-dimensional imbedding theorem in Sobolev space, then we can obtain the following.
Next, we give the corresponding error estimates. Choosing in the first equation of (2.8), then multiplying the three equations by , , , respectively, and adding them, we have Similarly, taking in the first equation of (3.8), then multiplying the three equations by , , , respectively, and adding them, we obtain Subtracting (3.11) from (2.10) and using (3.12), we get the error equation where . Let Then the error equation (3.13) can be rewritten as
Proof. Firstly, we estimate . Applying the Taylor formula, and noting the definitions of and in (2.10) and (3.8), respectively, we know so Notice that , (2.13) and (3.9), there exists a positive constant such that, when we have This implies (3.16) holds. Secondly, noting and (3.19), we obtain hence, by (2.16), (3.10), and we have This completes the proof of (3.17).
Proof. Expanding the term , we have For any , since and , then Let , thus . Applying transformation of variable, we know where By (3.28), we have On the other hand where Combining the above five equalities with (3.26), we can get Using the -inequality, we find Now we estimate . Noting (2.2), we have Furthermore, So When is sufficiently small, we can take suitable such that , from the above equalities we have Moreover, in view of and (3.9), we obtain Then Substituting the estimate of into (3.34), we have Since where lies between and , by (2.13), (3.9), and the inverse estimate, we obtain Thus, when we have , then From the assumption , we find that