Research Article  Open Access
FifthOrder Mapped SemiLagrangian Weighted Essentially Nonoscillatory Methods Near Certain Smooth Extrema
Abstract
Fifthorder mapped semiLagrangian weighted essentially nonoscillatory (WENO) methods at certain smooth extrema are developed in this study. The schemes contain the mapped semiLagrangian finite volume (MSLFV) WENO 5 method and the mapped compact semiLagrangian finite difference (MCSLFD) WENO 5 method. The weights in the more common scheme lose accuracy at certain smooth extrema. We introduce mapped weighting to handle the problem. In general, a cell average is applied to construct the MSLFV WENO 5 reconstruction, and the MCSLFD WENO 5 interpolation scheme is proposed based on an interpolation approach. An accuracy test and numerical examples are used to demonstrate that the two schemes reduce the loss of accuracy and improve the ability to capture discontinuities.
1. Introduction
The hyperbolic conservation laws are calculated in onedimensional form defined by where and can be either scalars or vectors. The fifthorder mapped semiLagrangian weighted essentially nonoscillatory (WENO) methods have been produced for the onedimensional scalar case and for systems near certain smooth extrema in this study. The scheme is divided into two parts: a mapped semiLagrangian finite volume (MSLFV) WENO method and a mapped compact semiLagrangian finite difference (MCSLFD) WENO method.
In [1], Henrick et al. proposed the mapped weighting near critical points. They contrasted the WENO 5 scheme and the mapped WENO 5 scheme and found that the WENO 5 method reached thirdorder accuracy at critical points, while the mapped weighted WENO 5 scheme achieved fifthorder accuracy. So the mapped WENO scheme reduced the loss of the accuracy and improved ability to capture discontinuities. Based on the theory of the mapped WENO scheme, Feng et al. used a piecewise polynomial function to propose the new mapped weights. The scheme reduced the influence of the discontinuities in the nonsmooth stencils, such that the underlying loss of accuracy was overcome [2]. In [3], Bryson and Levy designed a new method to solve the HamiltonJacobi equations using a mixed model consisting of mapped weights and a Godunovtype central method. This scheme reached highorder accuracy. Borges proposed a set of smoothness indicators and mapped weights for hyperbolic conservation laws. This scheme had less dissipation and achieved high accuracy, and they applied the method to twodimensional problems and found that it performed well [4].
In 1990, Lele proposed a series of compact methods, which were associated with a spectrallike solution [5]. They presented basic programmes for approximating the derivatives: a cellcentered compact scheme and midpoint interpolation. In [6], a highorder nonlinear scheme was proposed which considered a flux splitting and boundary scheme, followed by analysis of the asymptotic stability of weighted compact nonlinear scheme. Based on the WENO scheme, finite difference weighted compact programmes were developed, which associated the approximation of derivatives with the WENO scheme. In this way the method preserved the characteristics of normal compact methods and retained the ability to capture shock waves and discontinuities [7]. An efficient conservative compounded compact WENO scheme was proposed for shockturbulence interaction, analyzing the resolution properties, Fourier harmonies, and boundary closures. The boundary conditions affected the stability characteristics of the scheme, so explicit boundary conditions were proposed to deal with stability [8]. Zhang et al. developed a nonlinear weighted compact method that increased higherorder precision. They used a cellcentered compact approach, a splitting technique, and characteristic projection to increase the precision. The nonlinear compact WENO method was capable of capturing discontinuities without oscillation [9].
The hybrid model of the Lagrangian and Eulerian approaches produced the semiLagrangian approach, inheriting the advantages of the two approaches. This approach achieved high accuracy and allowed a weaker CFL condition. The LagrangianEulerian method proposed in 1974 solved a wide variety of timedependent multidimensional fluid problems. The methodology was stable and accurate [10]. A nonoscillatory Eulerian scheme was proposed to compute twodimensional Euler equations, level set equations, and equations of state. The level set function was used to hunt the interface in the scheme, so that the method was easily extendable to multidimensional and multilevel time integration [11]. A Lagrangian method with highorder ENO reconstruction was designed by Cheng and Shu for compressible Euler equations in which they compared figures of the four typical numerical fluxes in the Lagrangian scheme: Godunov flux, Dukowicz flux, LaxFriedrichs flux, and HartenLaxvan Leer contact wave flux [12]. In [13], Liu et al. combined the Lagrangian scheme with the LaxWendroff method to build a new scheme which was used to calculate the compressible Euler equations. The hybrid scheme was capable of saving computational cost.
In past years the semiLagrangian method has been very popular in transport planning [14–16]. Oscillationfree advection of interfaces, forwardtrajectory global scheme, and conservative and nonconservative forms were developed for transport schemes. Crouseilles applied the semiLagrangian method to Vlasov equations. This approach easily achieved highorder time accuracy and positivity [17, 18]. In [19–22], Qiu and Shu developed a series of semiLagrangian methods which allowed weaker CFL conditions and had stability, accuracy, and positivity properties.
In the present study, a mapped semiLagrangian WENO method is proposed for calculating hyperbolic conservation laws near certain smooth extrema. The method contains two schemes: the MSLFV WENO method and MCSLFD WENO method. The WENO scheme is widely used for hyperbolic conservation laws [23–26], but the WENO 5 scheme can only achieve thirdorder accuracy at certain smooth extrema. Mapped weighting has been designed to achieve an ideal order of accuracy at certain smooth extrema [1, 2]. The semiLagrangian finite volume WENO scheme uses cell averages to construct the WENO scheme [23, 24], and an interpolation approach is applied to build a semiLagrangian finite difference WENO scheme [9, 27]. In this study, the 3totalvariationdiminishing (TVD) RungeKutta (RK) scheme is applied to follow the backward characteristic line in a single time step for the case of variable characteristics [23, 28, 29]. As a result, the proposed schemes achieve fifthorder accuracy and overcome the potential loss of accuracy and also capture shock well.
The paper is organized as follows. The mapped finite volume WENO scheme for scalar hyperbolic conservation laws is reviewed at certain smooth extrema in Section 2. The MSLFV fifthorder WENO reconstruction scheme is given in Section 3. In Section 4, the compact semiLagrangian finite difference scheme for onedimensional hyperbolic conservation laws is analyzed and the fifthorder mapped weighting interpolation method is presented. In Section 5, an accuracy test and numerical tests are presented regarding fifthorder MSLFV and MCSLFD WENO methods for onedimensional hyperbolic conservation laws. The equations have certain smooth extrema, and the numerical results show that the proposed method works well in all cases. Concluding remarks and a perspective for future work are presented in Section 6.
2. A Review of Mapped FifthOrder Finite Volume WENO Scheme
The mapped fifthorder WENO scheme is shown as the solution of onedimensional scalar conservation laws, which are summarized by Putting the calculation region as and adopting the following spatial discretization: where , , we choose the center of the cell and cell length as , and .
For fifthorder WENO reconstruction, we first need to identify three thirdorder numerical fluxes. We set The fifthorder linear scheme is based on three thirdorder numerical fluxes , , and , respectively: The combination coefficients , , are termed linear weights.
It is noted that if the function is globally smooth, the linear weights , , are applied to obtain highorder accuracy. However, if the scheme oscillates near discontinuities, we need the assistance of the nonlinear weights proposed by Liu et al. to achieve highorder accuracy in smooth regions and capture the oscillations near discontinuities [24]. The nonlinear weights depend on the smoothness indicators which evaluate the smoothness of the functions , , and is given by These are The nonlinear weights are then determined by where are the linear weights and is a small and positive number to prevent the denominator becoming zero. In most of the numerical tests, .
To avoid the nonlinear weights producing any loss in precision at certain smooth extrema, Henrick et al. proposed the mapped weights [1] for conservation laws, which gave perfect precision of the WENO method at certain smooth extrema. The mapped weighted is defined as where and . The function has the following properties: The mapped nonlinear weights are then computed as In our numerical tests was taken as . The mappedweights method created highorder precision and reduced the loss of the accuracy at certain smooth extrema.
The mapped WENO of the flux is given by This is a mirror symmetric about of the abovementioned process for the approximations .
3. MSLFV FifthOrder WENO Reconstruction
The semiLagrangian finite volume method and mapped fifthorder WENO reconstruction scheme for scalar and system of conservation laws are presented in this section. We first show the semiLagrangian finite volume scheme and then introduce the mapped WENO 5 reconstruction.
3.1. SemiLagrangian Finite Volume Scheme for Scalar Case
Firstly, we integrate (2) to obtain the finite volume scheme The proposed semiLagrangian finite volume scheme is based on integrating (14); that is, we set up the integral in time such that Here, the threepoint Gaussian quadrature formula is used to approximate the integration in time, which limits the WENO reconstruction to fifthorder accuracy at most. From the abovementioned estimate, the corresponding equation is acquired as where , are the weights, and , , are the Gaussian quadrature points.
It should be noted that the , are not approximated by the WENO reconstruction directly, so we need to make use of the characteristic curves given by On the one hand, if is variable, and the thirdorder TVD RK method is applied to track the characteristic curves, we find the point at time level , which is built up by On the other hand, if is constant, the point can be built up by the formula Until now we have needed to substitute the above results of the reduction into (16) and evaluate the numerical flux at the cell point, so the semiLagrangian finite volume scheme is given by It is clear that the above result depends on the numerical flux at at time level . Lastly, we need to reconstruct the point value at at time level , which is described in detail in Section 3.3.
3.2. SemiLagrangian Finite Volume Scheme for System
Take the following onedimensional system of conservation laws in this section, defined by where To construct the semiLagrangian finite volume scheme for system (21), a similar derivation for the scalar case is applied to (21), as where , , , and , and is a LaxFriedrichs flux with
Note that the characteristics are defined by integrating the eigenvalues of for the case where the initial value problem of the system is to meet the characteristic curves The thirdorder TVD RK method is used to solve the abovementioned initial value problem. The solution is obtained from where . In a derivation similar to the scalar case, the semiLagrangian finite volume scheme for the system (21) is reduced to give
The WENO reconstruction is provided in detail in Section 3.3.
3.3. Mapped FifthOrder WENO Reconstruction
Ordinarily, fifthorder WENO reconstruction maintains fifthorder accuracy in smooth regions and captures discontinuities well but only achieves thirdorder accuracy at certain smooth extrema. Mapped nonlinear weights are used to better approximate the numerical flux at certain smooth extrema. In this subsection, we shall first consider the case when ; then we will consider the case when . The detail of the scheme has been omitted; only the necessary formulae are given in the following.
Case 1 (). In fifthorder reconstruction, we always use the small stencils , , and to construct the secondorder reconstruction polynomials . Then we use the large stencils to construct the fourthorder reconstruction polynomial . Defining we have the following reconstituted scheme: and the corresponding linear weights, indicated by , , in this case: The smoothness indicators take the following form: The nonlinear weights satisfy The mapped nonlinear weights are then computed from Here is to avoid the denominator becoming zero; we used the value in our numerical tests. The numerical flux is reconstructed by the following fifthorder mapped WENO scheme with the semiLagrangian finite volume scheme
Case 2 (). We use the the small stencils , , and to construct the secondorder reconstruction polynomials and then use the large stencils to derive a fourthorder reconstruction polynomial . We have and the corresponding linear weights, which are indicated by , , . In this case The smoothness indicators take the following form: The corresponding numeration is applied to obtain the following reconstituted scheme, so the numerical flux is reconstructed by the following fifthorder mapped WENO scheme and semiLagrangian finite volume scheme:
4. MCSLFD WENO Interpolation
The compact semiLagrangian finite difference method and mapped fifthorder WENO interpolation for scalar and system of conservation laws are given in this section. We first present the compact semiLagrangian finite difference scheme.
4.1. Compact SemiLagrangian Finite Difference Scheme for Scalar Case
From (2) with uniform grid , then at the grid point , a semidiscrete finite different scheme On the basis of (38) we propose a compact semiLagrangian finite difference scheme. Firstly, we set up the integral in time The threepoint Gaussian quadrature formula is then used to compute (39), which achieves fifthorder accuracy at most for the WENO reconstruction. From the above estimate, the corresponding equation is where , are the weights, and , , are the Gaussian quadrature points. In this study, we applied the sixthorder accuracy compact scheme to propose the compact semiLagrangian finite difference scheme. The sixthorder accuracy compact scheme is given as [5] where , ; ; and is the truncation error of the sixthorder tridiagonal scheme. By a corresponding derivation, the compact semiLagrangian finite difference scheme is given by From the above equation, we discover that (42) contains the corresponding compact coefficients and , which are determined by (41).
Until now, with the help of the characteristic curves, we have found the points and . These points are computed using identical arithmetic in the scalar case of the semiLagrangian finite volume scheme. Substituting the above results of the reduction into (42) and evaluating the numerical flux at the cell point, the compact semiLagrangian finite difference scheme is given by It is clear that the above result depends on the numerical flux at , at time level . Finally, we need to reconstruct the point value at at time level . This is described in detail in Section 4.3.
4.2. Compact SemiLagrangian Finite Difference Scheme for System
Parallel to the system in the semiLagrangian finite volume scheme, the characteristics are defined by integrating the eigenvalues of , denoted by , . In the same way, the thirdorder TVD RK method is used to solve the characteristic curves. The solution is given by where . The semiLagrangian finite different scheme for the system (21) is taken as where and are the corresponding compact coefficients determined by (41). The WENO interpolation is provided in detail in Section 4.3.
4.3. Mapped Fifth Order WENO Interpolation
In the subsection, we shall first consider the case when , after that we will consider the case when . The details of the method are omitted and necessary formulas are given in the following.
Case 1 (). In the fifth order interpolation, we always use the three points stencils , , and to interpolate polynomials . Then we use the stencil to interpolate polynomial . We have the following scheme: and , , , are the corresponding linear weights given by The smoothness indicators are taken on the following form: The nonlinear weights satisfy The mapped nonlinear weights are then computed from Here is to avoid the denominator becoming zero; we use the value in our numerical tests. The numerical flux is computed by the following fifth order mapped WENO scheme with the semiLagrangian compact finite difference scheme:
Case 2 (). We use the three points stencils , , and and the stencil to build the scheme. We have and the corresponding linear weights, which are denoted by , , . In this case, The smoothness indicators are taken on the following form: The numerical flux is reconstructed by the following fifth order mapped WENO scheme:
4.4. Boundary Conditions
It should be pointed out that the derivedboundary and nearboundary schemes are significant for the fifthorder mapped MCSLFD WENO schemes. The boundary conditions are crucial for the order of accuracy, and they affect the stability characteristics of the scheme. In this study, the fourthorder accuracy boundary programme of Zhang et al. [9] was used as follows: It should be observed that the accuracy of the boundary conditions is fourthorder, but that does not reduce the fifthorder accuracy of the MCSLFD WENO schemes; that is, fourthorder accuracy boundary conditions maintain fifthorder accuracy and stability of the scheme proposed in this study.
Remark 1. Negative linear weights in a WENO scheme produce oscillations and instability. Shi et al. [30] have dealt with this problem using a simple and effective splitting technique which obviates the need to remove the negative weights and retains the stability and accuracy of the scheme.
Remark 2. If the systems under consideration have characteristic variables, such as Ii and Xiao [28] have produced a method incorporating a decoupled system of characteristic variables and Riemann invariants and then solved the linear systems for primitive variables along the characteristic curves, which is the approach adopted in this paper.
Remark 3. When the CFL number is large enough to make , Qiu and Shu have proposed a large timestep evolution method to handle this condition [19].
5. Numerical Examples
Massive numerical experiments to assess the performance of the MSLFV WENO 5 and MCSLFD WENO 5 schemes are described in this section. We present the results of our numerical tests for scalar and for system of conservation laws. If in the following examples the characteristics were variable, we adopted the thirdorder TVD RK method. The mapped semiLagrangian WENO 5 schemes and semiLagrangian WENO 5 schemes allow a weaker CFL condition. In the examples, the CFL number is taken as 5.9 for the linear advection problem and Burger's equation and 9.9 for the onedimensional Euler equations. A CFL number of 0.2 is chosen for the FV WENO 5 and FD WENO 5 schemes. All solutions are computed using uniform meshes. In all of the simulations, the mapped semiLagrangian schemes use . For the semiLagrangian, FV and FD WENO schemes use .
5.1. Accuracy Test
Example 1 (the linear advection problem). We check the order of accuracy of MSLFV, MCSLFD, SLFV, CSLFD, FV, and FD WENO 5 methods for the linear advection problem The initial condition is given by , the investigative domain is , and the boundary conditions are periodic. The exact solution of this problem is given by The norm of the error is computed at time and . The errors and numerical orders of accuracy for the MSLFV WENO 5 scheme and MCSLFD WENO 5 scheme are shown in Table 1. The convergence results of the scheme without the mapped weights are listed in Table 2. The convergence rates of Tables 1 and 2 show that MSLFV WENO 5 and MCSLFD WENO 5 with schemes can reach fifthorder accuracy. In fact, the error is thirdorder accuracy in the norm with as shown in Table 2. The and errors of the FV and FD WENO 5 methods are shown in Table 3. Tables 1 and 3 show the orders of the accuracy for the MSLFV, MCSLFD, FV, and FD WENO 5 methods. We see that the FV and FD WENO 5 methods do not achieve fifthorder accuracy for the linear advection problem. It is important to point out that the FV and FD methods are worse than the MSLFV and MCSLFD methods for the linear advection problem. So we can confirm that the mapped weights scheme gives a good approximation to the exact solution, and the MSLFV WENO 5 and MCSLFD WENO 5 methods perform well for the linear advection problem.


