The interpolation of piecewise smooth function from the discrete sample points is always an important problem of numerical computation. The simple linear Lagrangian interpolation is suitable for a set of smooth sample points and achieves the optimal accuracy. However, we often need to interpolate a set of discrete points which contains corner (or jump) discontinuities. In these cases, the Gibbs oscillations generated by linear Lagrangian interpolations around the discontinuities will decrease the numerical accuracy. Then some preferred nonlinear algorithms should be used to obtain stable numerical approximation. The nonlinear essentially non-oscillatory (ENO) interpolation, which can tackle this problem well, was first presented in [1][2] for solving hyperbolic conservation laws. To eliminate the effect of non-smooth parts of sample points, the ENO algorithm adaptively chooses the smoothest stencil from the candidates to reconstruct the piecewise function. The indicator of smoothness of each stencil is defined as the Newton divided difference of sample points on this stencil. More information about ENO algorithm can be found in [3]-[5].

In the procedure of ENO interpolation, while the smoothness of all the candidate stencils is measured, only the smoothest one is finally conserved even though all the stencils are smooth enough to be used. To gather all the potential stencils, in [6], the authors presented the weighted ENO (WENO) algorithm based on ENO. The aim of WENO algorithm is to keep the ENO property in non-smooth regions and meanwhile improve the order of accuracy in smooth regions. To obtain it, in the WENO algorithm, each candidate stencil is assigned a nonlinear weight which measures the contribution of this stencil for the final convex combination. The nonlinear weights are devised to approach linear optimal weights in smooth region to improve the accuracy of interpolation. While there exist stencils across the discontinuities, the weights assigned to these non-smooth stencils can be ignored almost. In [7], the authors showed new indicators of smoothness which emulate the idea of minimizing the total variation of function. The new indicators of smoothness are defined as the L 2 norm of the derivatives of the interpolation polynomials. The indicators of smoothness in [7] ensure the optimal order of accuracy in smooth region and ENO property when there exist stencils that contain the discontinuities. About the improvements and applications of WENO algorithm, the interested reader can refer to [8]-[16].

Around the discontinuities, the classical WENO algorithm [7] generally only achieves the same order of accuracy as the corresponding ENO interpolation. It was shown in [7] for reconstructing the numerical flux at cell boundary based on the cell averages. What leads to the degeneration of order of accuracy is the classical WENO cannot distribute reasonable optimal weights to the smooth stencils when the discontinuity appears. In [17], the authors proposed a power WENO algorithm to improve the accuracy near discontinuities. However, it does not obtain the maximum theoretical order of accuracy. Then, the authors in [18] succeed in obtaining the maximum theoretical accuracy close to the discontinuities by an improved WENO algorithm (WENO-AW). Unlike the fixed linear optimal weights proposed in [7][19], they devise nonlinear optimal weights to tackle this problem. The nonlinear optimal weights are expressed as the nonlinear convex combination of three vectors of linear optimal weights. Each vector of linear optimal weights is appropriate for one special case. The proofs of maximum theoretical accuracy near discontinuities and ENO property were presented in [18]. They also give another algorithm to raise the accuracy of WENO algorithm for the interval which contains the corner discontinuity. In [17][18][20], certain modified indicators of smoothness are used to detect the corner and jump discontinuities. At the same time, these indicators of smoothness conserve the optimal accuracy and ENO property. In this paper, we only consider how to improve the order of accuracy near discontinuities by using a more efficient algorithm. If one wants to obtain the optimal accuracy in the interval containing the discontinuity, the algorithm in [18] will be a commendable choice.

As mentioned above, the algorithm in [18] recovers the optimal accuracy near discontinuities by devising a set of nonlinear optimal weights. However, the new nonlinear optimal weights are computed by an extra WENO algorithm. And in this procedure, the indicators of smoothness of the bigger stencils also need to be calculated. As shown in [18], the computational costs of new WENO algorithm are more than double when compared with the classical WENO. In this paper, we present a simple WENO algorithm to recover the optimal accuracy near discontinuities. In order to describe this algorithm clearly, we take the 6th-order WENO algorithm for example. To ensure the optimal accuracy near discontinuities, the classical WENO algorithm is divided into three sub-WENO procedures. We first construct two 5th-order WENO approximations by using the former two 4-points stencils and latter two 4-points stencils, respectively. In the next, we construct the 6th-order WENO approximation by using two 5-points stencils and corresponding 5th-order WENO approximations which have been obtained. From the statement above, we need three sub-WENO procedures in this algorithm and in each sub-WENO procedure we only need information of two stencils. Unlike the classical WENO, which directly constructs the 6th-order approximation by the nonlinear convex combination of three 4th-order approximations, the order of accuracy of new WENO algorithm grows one by one and this nonlinear interpolation truly confirms the optimal accuracy near discontinuities. Since the 5-points stencils are used in this procedure, we have to compute their indicators of smoothness. To control the oscillations and reduce the computational cost, we express the indicators of smoothness of 5-points stencils as the product of two corresponding 4-points substencils. The new WENO algorithm here can be reformulated into the similar compact form as the classical WENO algorithm and not much computation is introduced. Furthermore, it is also easy to extend the algorithm to the cases of higher order.

The organization of this paper is as follows. In Section 2, we review the classical WENO [7] and WENO-AW [18] algorithms. In Section 3, we show the new WENO interpolation and prove the statement of optimal order of accuracy near discontinuity and property of ENO. In Section 5, we test the numerical accuracy and computational costs.

In this section, we review the classical WENO [7] and WENO-AW algorithm [18] in point values. The interested reader can also refer to [21][22] for a full statement of WENO interpolations.

In this section, a simple 6th-order sub-WENO algorithm is presented, which has the same aim as the WENO-AW algorithm proposed in [18], to achieve the optimal order near the discontinuities. The second algorithm in [18] is to improve the order of accuracy of when the middle interval ( x j − 1 , x j ) contains the discontinuity. As shown in Section 2.2, the extra three indicators of smoothness of big stencils β 1 5 , β 2 5 and β 2 6 are required to calculate the adapted nonlinear optimal weights. The computational cost of the WENO-AW algorithm in [18] is more than double when compared with the classical WENO algorithm [7]. In the classical 6th-order WENO algorithm, the idea is to combine three 4th-order linear interpolations to achieve 6th-order accuracy at smooth regions and the potential 5th-order interpolations are skipped. It is the probable reason which leads to the degeneration of algorithm near discontinuities. To recover the accuracy near discontinuities, we divide the classical WENO into several sub-WENO procedures. In each sub-WENO procedure, only two stencils are used. The order of accuracy grows by only one at most by combining two stencils and the ENO property is also conserved (Figure 1).

Figure 1. The left is the tree structure of the 6th-order sub-WENO algorithm. The right is the tree structure of the 6th-order classical WENO algorithm.

In the 6th-order sub-WENO algorithm, there are three 4-points stencils can be used and there will be three sub-WENO procedures as shown in Section 3. In the first sub-WENO procedure denoted by { S 0 4 , S 1 4 } ↪ S 1 5 , emulating the classical WENO in Section 2, we use the stencils S 0 4 and S 1 4 to compute the 5th-order approximation at x j − 1 2 ,

We extract the formulas of interpolation polynomials p i 4 ( x ) ( i = 0 , 1 ) at x j − 1 2 from (1),

The nonlinear weights are computed similarly as (4) by

and the unnormalized weights are defined as

The linear optimal weights in (15) are chose to be d 0 4 = 3 8 and d 1 4 = 5 8 to satisfy

The indicators of smoothness β i 4 ( i = 0 , 1 ) are extracted from (3), and we repeat them for completeness of new algorithm,

Up to now, the first sub-WENO procedure { S 0 4 , S 1 4 } ↪ S 1 5 is completed by emulating the classical WENO algorithm. This sub-WENO procedure can achieve 5th-order approximation to f j − 1 2 at smooth region and keep the ENO property when there is one non-smooth stencil. Actually, when there is one non-smooth stencil, this sub-WENO procedure still ensure the 4th-order of accuracy. The proofs of assertions are ignored here, since the properties of the algorithm after integrating three sub-WENO procedures are our aims.

In the second sub-WENO procedure { S 1 4 , S 2 4 } ↪ S 2 5 , we use the stencils S 1 4 and S 2 4 to obtain another 5th-order approximation to f j − 1 2 ,

where the values of two 4th-order interpolation polynomials at x j − 1 2 are

The nonlinear weights in (17) are computed as

and the unnormalized weights are defined as

The linear optimal weights are chose to be d 1 4 = 5 8 and d 2 4 = 3 8 for satisfying

The indicators of smoothness β i 4 ( i = 1 , 2 ) are also chosen from (3),

The second sub-WENO procedure { S 1 4 , S 2 4 } ↪ S 2 5 is similar to the first one but the nonlinear interpolation is operated on the stencils S 1 4 and S 2 4 . It is noted that in the two sub-WENO procedures above, we use the several same notations, such as the nonlinear weight w 1 4 in (14) and (19). But these same notations are independent in the different sub-WENO procedures.

Through the former sub-WENO procedures, we arrive the 5th-order approximations satisfying

and

on the 5-points stencils S 1 5 and S 2 5 , respectively, when discretized data is smooth. In the third sub-WENO procedure { S 1 5 , S 2 5 } ↪ S 2 6 , we arrange to obtain the 6th-order approximation to f j − 1 2 by combination of F 1 5 ( x j − 1 2 ) and F 2 5 ( x j − 1 2 ) ,

The nonlinear weights are defined as

where the unnormalized weights are defined as

The optimal weights are chosen to be d 1 5 = 1 2 and d 2 5 = 1 2 for satisfying

When the discontinuity lies in ( x j + 1 , x j + 2 ) or ( x j − 3 , x j − 2 ) , the choice of (2) is valid and the final algorithm can achieve 5th-order accuracy and keep the ENO property. However, if there exists a discontinuity appears in ( x j , x j + 1 ) or ( x j − 2 , x j − 1 ) , then S 1 5 and S 2 5 are both non-smooth. And in result, both stencils are distributed comparative weights and the oscillation will be inevitable. To approach the optimal accuracy and preserve the ENO property, the effective choices are

Taking S 1 5 for example, since S 1 5 = S 0 4 ∪ S 1 4 , we express the indicator of smoothness parent stencil S 1 5 as the product of substencils S 0 4 and S 1 4 . This option will be extended to the higher order sub-WENO algorithms. If the discontinuity appears in ( x j , x j + 1 ) or ( x j + 1 , x j + 2 ) , by using the indicators of smoothness in (25), the contribution of stencil S 2 5 can be ignored and S 1 5 will dominate the final combination. The similar result can be obtained when the discontinuity lies in ( x j − 3 , x j − 2 ) or ( x j − 2 , x j − 1 ) . In addition, the indicators of smoothness β i 4 ( i = 0 , 1 , 2 ) have been obtained in the former sub-WENO procedures and we do not need extra computational cost. For the simplification of the equations, we replace ϵ + β i 5 ( i = 1 , 2 ) in denominators of (24) by ( ϵ + β i − 1 4 ) ( ϵ + β i 4 ) ( i = 1 , 2 ) , respectively, and (24) becomes

The common term ϵ + β 1 4 will be cancelled when we substitute the unnormalized weights into (23). So, we actually define the indicators of smoothness β 1 5 and β 2 5 as

For the higher order sub-WENO algorithms, the choice of indicators of smoothness is presented in Section 4 in detail. The proofs of achieving optimal accuracy and keeping ENO property will be shown in theorem 1. Before giving this theorem, we first simplify the sub-WENO algorithm which will facilitate the proof of theorem 1.

Inserting the weights (14) into (12) gives

and similarly inserting the weights (19) into (17) gives

Finally, substituting the 5th-order approximations (28) (29) into (22), we obtain

Through reorganization, the nonlinear weights in (30) are

where the unnormalized weights are

Theorem1.Supposethatthestencil S 2 6 containsadiscontinuityatmost, theexponentinunnormalizedweights q ≥ 1 and ϵ ≤ O ( Δ x 4 ) .Thenthe 6th-ordersub-WENOalgorithm (30) (31) (32) satisfiesthefollowingthreecases:

Case 1: if the stencil S 2 6 is smooth, then

Case 2: if there is a discontinuity lies in ( x j − 3 , x j − 2 ) , then

or if there is a discontinuity lies in ( x j + 1 , x j + 2 ) , then

Case 3: if there is a discontinuity lies in ( x j − 2 , x j − 1 ) , then

or if there is a discontinuity lies in ( x j , x j + 1 ) , then