The Backward Euler Fully Discrete Finite Volume Method for the Problem of Purely Longitudinal Motion of a Homogeneous Bar

and Applied Analysis 3 for all uh x ∈ Uh, where φi x is the basis function associated with the nodes xi i 1, 2, . . . , r − 1 , φi x ⎧ ⎪ ⎪⎨ ⎪ ⎪⎩ 1 − xi − x hi , x ∈ Ii, 1 − x − xi hi 1 , x ∈ Ii 1, 0, x / ∈ Ii ∪ Ii 1. 2.1 It is easy to know that the derivative of uh with respect to x is uhx x uh xi − uh xi−1 hi , xi−1 ≤ x ≤ xi, i 1, 2, . . . , r. 2.2 The test function space Vh ⊂ L2 I associated with the dual partition T ∗ h is defined as the set of all piecewise constants with vh 0 vh 1 0 for all vh x ∈ Vh. We may choose the basis function ψj x of Vh in such a way that ψj x is the characteristic function of I∗ j , that is, ψj x { 1, x ∈ I∗ j , 0, x / ∈ I∗ j , j 1, 2, . . . , r − 1. 2.3 Then for any vh x ∈ Vh can be expressed as vh x ∑r−1 j 1 vh xj ψj x . Obviously, Uh ⊂ H1 0 I , Vh ⊂ L2 I , dimUh dimVh r − 1. 2.4 Meanwhile, Uh ⊂ W1,∞ 0 I . Thirdly, for the time interval 0, T , we give an isometric partition and denote the nodes ti iτ , i 0, 1, . . . ,N, τ T/N. We introduce some notations for functions u x, t and f ux x, t : u u x, tn , uj u ( xj , t ) , unj u ( xj , tn ) , ∂tu n u 1 − u τ , ∂ttu n u 1 − 2u un−1 τ2 , u 1/2 u 1 u 2 , un,1/4 u 1 2u un−1 4 , f1/2 ux f ( u 1 x ) f ux 2 , f1/4 ux f ( u 1 x ) 2f ux f ( un−1 x ) 4 , f 1/4 ∗ ux 3f ux f ( un−1 x ) 4 . 2.5 4 Abstract and Applied Analysis Then we can get ∂ttu n ∂tu n − ∂tun−1 τ , un,1/4 u 1/2 un−1/2 2 , 1 2 ( ∂tu n ∂tun−1 ) 1 τ ( u 1/2 − un−1/2 ) . 2.6 Let u be the solution of 1.1 . Integrating 1.1 a over the dual element I∗ j ∈ T ∗ h , we obtain ∫xj 1/2 xj−1/2 uttdx uxt ( xj−1/2 ) − uxt ( xj 1/2 ) f ( ux ( xj−1/2 )) − fux ( xj 1/2 )) 0, 2.7 where j 0, 1, . . . , r, x−1/2 x0, xr 1/2 xr , and 0 < t ≤ T . The problem 2.7 can be rewritten in a variational form. For any arbitrary vh ∈ Vh, we multiply the integral relation in 2.7 by vh xj and sum over all j 0, 1, . . . , r to obtain utt, vh a∗ ut, vh b∗ ( f u , vh ) 0, ∀vh ∈ Vh, t ∈ 0, T , u 0 u0, ut 0 u1, 2.8 where for any arbitrary w ∑r−1 j 1 wjψj ∈ Vh, the bilinear forms a∗ v,w , b∗ f v , w are defined by