Abstract
Fifth-order mapped semi-Lagrangian weighted essentially nonoscillatory (WENO) methods at certain smooth extrema are developed in this study. The schemes contain the mapped semi-Lagrangian finite volume (M-SL-FV) WENO 5 method and the mapped compact semi-Lagrangian finite difference (M-C-SL-FD) 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 M-SL-FV WENO 5 reconstruction, and the M-C-SL-FD 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 one-dimensional form defined by where and can be either scalars or vectors. The fifth-order mapped semi-Lagrangian weighted essentially nonoscillatory (WENO) methods have been produced for the one-dimensional scalar case and for systems near certain smooth extrema in this study. The scheme is divided into two parts: a mapped semi-Lagrangian finite volume (M-SL-FV) WENO method and a mapped compact semi-Lagrangian finite difference (M-C-SL-FD) 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 third-order accuracy at critical points, while the mapped weighted WENO 5 scheme achieved fifth-order 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 Hamilton-Jacobi equations using a mixed model consisting of mapped weights and a Godunov-type central method. This scheme reached high-order 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 two-dimensional problems and found that it performed well [4].
In 1990, Lele proposed a series of compact methods, which were associated with a spectral-like solution [5]. They presented basic programmes for approximating the derivatives: a cell-centered compact scheme and midpoint interpolation. In [6], a high-order 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 shock-turbulence 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 higher-order precision. They used a cell-centered 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 semi-Lagrangian approach, inheriting the advantages of the two approaches. This approach achieved high accuracy and allowed a weaker CFL condition. The Lagrangian-Eulerian method proposed in 1974 solved a wide variety of time-dependent multidimensional fluid problems. The methodology was stable and accurate [10]. A nonoscillatory Eulerian scheme was proposed to compute two-dimensional 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 high-order 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, Lax-Friedrichs flux, and Harten-Lax-van Leer contact wave flux [12]. In [13], Liu et al. combined the Lagrangian scheme with the Lax-Wendroff 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 semi-Lagrangian method has been very popular in transport planning [14–16]. Oscillation-free advection of interfaces, forward-trajectory global scheme, and conservative and nonconservative forms were developed for transport schemes. Crouseilles applied the semi-Lagrangian method to Vlasov equations. This approach easily achieved high-order time accuracy and positivity [17, 18]. In [19–22], Qiu and Shu developed a series of semi-Lagrangian methods which allowed weaker CFL conditions and had stability, accuracy, and positivity properties.
In the present study, a mapped semi-Lagrangian WENO method is proposed for calculating hyperbolic conservation laws near certain smooth extrema. The method contains two schemes: the M-SL-FV WENO method and M-C-SL-FD WENO method. The WENO scheme is widely used for hyperbolic conservation laws [23–26], but the WENO 5 scheme can only achieve third-order accuracy at certain smooth extrema. Mapped weighting has been designed to achieve an ideal order of accuracy at certain smooth extrema [1, 2]. The semi-Lagrangian finite volume WENO scheme uses cell averages to construct the WENO scheme [23, 24], and an interpolation approach is applied to build a semi-Lagrangian finite difference WENO scheme [9, 27]. In this study, the 3-total-variation-diminishing (TVD) Runge-Kutta (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 fifth-order 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 M-SL-FV fifth-order WENO reconstruction scheme is given in Section 3. In Section 4, the compact semi-Lagrangian finite difference scheme for one-dimensional hyperbolic conservation laws is analyzed and the fifth-order mapped weighting interpolation method is presented. In Section 5, an accuracy test and numerical tests are presented regarding fifth-order M-SL-FV and M-C-SL-FD WENO methods for one-dimensional 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 Fifth-Order Finite Volume WENO Scheme
The mapped fifth-order WENO scheme is shown as the solution of one-dimensional 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 fifth-order WENO reconstruction, we first need to identify three third-order numerical fluxes. We set The fifth-order linear scheme is based on three third-order 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 high-order accuracy. However, if the scheme oscillates near discontinuities, we need the assistance of the nonlinear weights proposed by Liu et al. to achieve high-order 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 mapped-weights method created high-order 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. M-SL-FV Fifth-Order WENO Reconstruction
The semi-Lagrangian finite volume method and mapped fifth-order WENO reconstruction scheme for scalar and system of conservation laws are presented in this section. We first show the semi-Lagrangian finite volume scheme and then introduce the mapped WENO 5 reconstruction.
3.1. Semi-Lagrangian Finite Volume Scheme for Scalar Case
Firstly, we integrate (2) to obtain the finite volume scheme The proposed semi-Lagrangian finite volume scheme is based on integrating (14); that is, we set up the integral in time such that Here, the three-point Gaussian quadrature formula is used to approximate the integration in time, which limits the WENO reconstruction to fifth-order accuracy at most. From the above-mentioned 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 third-order 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 semi-Lagrangian 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. Semi-Lagrangian Finite Volume Scheme for System
Take the following one-dimensional system of conservation laws in this section, defined by where To construct the semi-Lagrangian finite volume scheme for system (21), a similar derivation for the scalar case is applied to (21), as where , , , and , and is a Lax-Friedrichs 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 third-order 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 semi-Lagrangian 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 Fifth-Order WENO Reconstruction
Ordinarily, fifth-order WENO reconstruction maintains fifth-order accuracy in smooth regions and captures discontinuities well but only achieves third-order 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 fifth-order reconstruction, we always use the small stencils , , and to construct the second-order reconstruction polynomials . Then we use the large stencils to construct the fourth-order 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 fifth-order mapped WENO scheme with the semi-Lagrangian finite volume scheme
Case 2 (). We use the the small stencils , , and to construct the second-order reconstruction polynomials and then use the large stencils to derive a fourth-order 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 fifth-order mapped WENO scheme and semi-Lagrangian finite volume scheme:
4. M-C-SL-FD WENO Interpolation
The compact semi-Lagrangian finite difference method and mapped fifth-order WENO interpolation for scalar and system of conservation laws are given in this section. We first present the compact semi-Lagrangian finite difference scheme.
4.1. Compact Semi-Lagrangian 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 semi-Lagrangian finite difference scheme. Firstly, we set up the integral in time The three-point Gaussian quadrature formula is then used to compute (39), which achieves fifth-order 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 sixth-order accuracy compact scheme to propose the compact semi-Lagrangian finite difference scheme. The sixth-order accuracy compact scheme is given as [5] where , ; ; and is the truncation error of the sixth-order tridiagonal scheme. By a corresponding derivation, the compact semi-Lagrangian 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 semi-Lagrangian finite volume scheme. Substituting the above results of the reduction into (42) and evaluating the numerical flux at the cell point, the compact semi-Lagrangian 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 Semi-Lagrangian Finite Difference Scheme for System
Parallel to the system in the semi-Lagrangian finite volume scheme, the characteristics are defined by integrating the eigenvalues of , denoted by , . In the same way, the third-order TVD RK method is used to solve the characteristic curves. The solution is given by where . The semi-Lagrangian 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 semi-Lagrangian 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 derived-boundary and near-boundary schemes are significant for the fifth-order mapped M-C-SL-FD 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 fourth-order accuracy boundary programme of Zhang et al. [9] was used as follows: It should be observed that the accuracy of the boundary conditions is fourth-order, but that does not reduce the fifth-order accuracy of the M-C-SL-FD WENO schemes; that is, fourth-order accuracy boundary conditions maintain fifth-order 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 time-step evolution method to handle this condition [19].
5. Numerical Examples
Massive numerical experiments to assess the performance of the M-SL-FV WENO 5 and M-C-SL-FD 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 third-order TVD RK method. The mapped semi-Lagrangian WENO 5 schemes and semi-Lagrangian 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 one-dimensional 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 semi-Lagrangian schemes use . For the semi-Lagrangian, FV and FD WENO schemes use .
5.1. Accuracy Test
Example 1 (the linear advection problem). We check the order of accuracy of M-SL-FV, M-C-SL-FD, SL-FV, C-SL-FD, 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 M-SL-FV WENO 5 scheme and M-C-SL-FD 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 M-SL-FV WENO 5 and M-C-SL-FD WENO 5 with schemes can reach fifth-order accuracy. In fact, the error is third-order 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 M-SL-FV, M-C-SL-FD, FV, and FD WENO 5 methods. We see that the FV and FD WENO 5 methods do not achieve fifth-order accuracy for the linear advection problem. It is important to point out that the FV and FD methods are worse than the M-SL-FV and M-C-SL-FD 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 M-SL-FV WENO 5 and M-C-SL-FD WENO 5 methods perform well for the linear advection problem.
5.2. Numerical Tests
Example 2 (Burgers equation). In order to demonstrate the stability of the M-SL-FV WENO 5 scheme and M-C-SL-FD WENO 5 scheme, the nonlinear scalar Burgers equation is We take as the initial condition. The domain of the equation in this study is taken as , and with periodic boundary conditions. We show the results at and with . The solution remains smooth at , but a shock wave is produced at . The outcomes of the M-SL-FV WENO 5 and M-C-SL-FD WENO 5 schemes at time and are shown in Figures 1 and 2, respectively. The results are seen to be reasonable; both the M-SL-FV WENO 5 and M-C-SL-FD WENO 5 schemes solve the numerical oscillations very well at .
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Now, we begin to study the one-dimensional Euler system, written as where Here , , , and are the density, velocity, total energy, and pressure, respectively. The equation of state has the following form: and the ratio of specific heat .
Example 3 (the Sod problem). The sod problem is the Euler equation (61), specified by the following initial conditions: Setting as the computational domain of this problem, we show the results at and with periodic boundary conditions. In Figure 3, the outcomes of the M-SL-FV WENO 5 and M-C-SL-FD WENO 5 schemes with a uniform grid of 200 points are shown. It is important to point out that the shock and contact discontinuity are correctly captured by both schemes.
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Example 4 (The Lax problem). We consider the lax problem, which is the Euler equation (61) and has the following starting conditions: Setting as the investigative domain of this problem, the solution is up to and with periodic boundary conditions. In the example, the M-SL-FV WENO 5 scheme and M-C-SL-FD WENO 5 scheme are used to calculate the solution. The simulated density of the lax problem with a uniform gird of 200 points against the exact solution are displayed in Figure 4. It can be seen that the two schemes maintain the ENO property and have a good nonoscillatory resolution at the discontinuities.
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Example 5 (the shock-turbulence interaction problem). The shock-turbulence interaction problem is the Euler equation (61), specified by the following initial conditions: Here we take . Setting as the computational domain of this problem, the density is plotted at and with periodic boundary conditions. The outcomes of the M-SL-FV WENO 5 scheme and M-C-SL-FD WENO 5 scheme are displayed in Figure 5. The figure shows that the two schemes can capture sharp, nonoscillatory shock near discontinuities.
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Example 6 (the two interacting blast waves problem). In this example, we consider two interacting blast waves, which is defined by the Euler equation (61) and has the following starting conditions: Setting as the investigative domain of this problem, the solution is up to and with periodic boundary conditions. In the example, the simulated density of the two interacting blast waves problem are displayed for the M-SL-FV WENO 5 and M-C-SL-FD WENO 5 schemes in Figure 6 for a mesh of 400 cells. The figure shows clearly that the strong shock waves and contact discontinuities are best resolved by the M-SL-FV WENO 5 and M-C-SL-FD WENO 5 schemes. The numerical solutions are very satisfactory.
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6. Concluding Remarks
Mapped semi-Lagrangian WENO methods are proposed in this paper. The M-SL-FV WENO 5 and M-C-SL-FD WENO 5 schemes were applied to one-dimensional hyperbolic conservation laws. Both schemes permit weaker CFL conditions, at the same time achieving fifth-order accuracy at certain smooth extrema and nonoscillatory shock near discontinuities. The numerical examples using the two schemes have demonstrated that they improve both accuracy and the ability to capture shock. The study of mapped semi-Lagrangian WENO schemes will be further expanded into the challenging areas of multidimensional nonlinear problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by the National Science Foundation of China (11271100 and 11301113), the Ph.D. Programs Foundation of Ministry of Education of China (no. 20132302120057), the class General Financial Grant from the China Postdoctoral Science Foundation (Grant no. 2012M510933), the Fundamental Research Funds for the Central Universities (Grant nos. HIT. NSRIF. 2011003 and HIT. A. 201412), the Program for Innovation Research of Science in Harbin Institute of Technology (Grant no. PIRS OF HIT A201403), and Harbin Science and Technology Innovative Talents Project of Special Fund (2013RFXYJ044).