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 Open Access
Higherorder finite volume method with semiLagrangian scheme for onedimensional conservation laws
 Lang Wu^{1},
 Songsong Li^{1, 2} and
 Boying Wu^{1}Email author
https://doi.org/10.1186/s136620140353y
© Wu et al.; licensee Springer 2015
 Received: 5 September 2014
 Accepted: 26 December 2014
 Published: 21 March 2015
Abstract
In this paper, a highorder, semiLagrangian finite volume (SLFV) method based on the WENO approach is proposed in order to manage onedimensional conservation laws. The proposed method successfully integrates WENO reconstructions and the semiLagrangian method. More specifically, the Taylor expansion of time is used to approximate the time integration, deployed to boost temporal accuracy. Next, characteristic curves are applied to replace the time level by points in the semiLagrangian method. The value of these points can then be reconstructed by WENO schemes to increase their accuracy in space. Both highorder accuracies in space and time, respectively, are achieved. Moreover, computational experiments allow for a weaker CFL condition, provided in detail to validate the performance of the proposed SLFVbased WENO method.
Keywords
 semiLagrangian method
 WENO reconstructions
 Taylor expansion
 Euler system
1 Introduction
The LagrangianEulerian formulation with moving grids was applied to unsteady, compressible Euler equations, under a conservative scheme [5]. The Eulerian method was able to manage the strongly nonlinear processes without introducing additional complexity. The Lagrangian method was developed in FV, where the scheme essentially combined with the nonoscillatory (ENO) method to solve the Euler equations [6]. The scheme obtained highorder precision with Cartesian and cylindrical coordinates. Liu et al. developed a Lagrangian method with a Taylor expansion in time, called the LaxWendroff method, developed in order to solve compressible Euler equations and fulfill planned accuracy in compact spatial stencils [7]. In another study [8, 9], the Lagrangian scheme was used to determine a solution for compressible fluid flows in cylindrical coordinates, proven to maintain symmetry and conservation characteristics. As for columniform problems, by way of obtaining a method which preserves circular harmony and conservative characteristics, the momentum equation was discretized alongside the local polar and angular directions.
In yet another relevant study [10], Cheng and Knorr proposed an SL method, to considerable interest of computational scientists. According to the property of characteristic curves, the SL method calculated the value of points which end at the grid point backward in time. Over the last few years, a semiLagrangian methodology has been explored at length for its ability to solve the Vlasov equation [11, 12] and transport scheme [13–15]. The semiLagrangian method can be divided into two categories: the backward semiLagrangian (BSL) method, and forward semiLagrangian (FSL) method. The BSL method utilizes the point value with \(t^{n+1}\), where the value point \(t^{n}\) must be identified; the FSL method functions in a manner exactly opposite. Both semiLagrangian methods utilized Lagrange reconstruction, spline reconstruction, and Hermite reconstruction to compute their characteristic variables [11].
The semiLagrangian scheme maintains uniform highorder precision and nonoscillatory shock transitions near discontinuities, which simultaneously allowed weaker CFL conditions to ensure computational efficiency. SemiLagrangian methods have been applied more recently to FV structures, and to discontinuous Galerkin and FD schemes [16–19]. Specifically, in one recent study [18, 19], Qiu and Shu developed an SL finite difference method using WENO approaches to approximate the advection equation and the Vlasov equation. The FD WENO approach used point values instead of cell averages for the Strang split scheme. A positivity preservation of the semiLagrangian discontinuous Galerkin method was also proposed [20], which demonstrated overall consistency and preciseness. Another study [16] proposed a maximum principle preserving the semiLagrangian discontinuous Galerkin method, and it further developed a positivity preserving limiter.
Highorder SLFV approaches are created to approximate scalars and Euler equations by combining the advantages of WENO reconstructions [21–25] and the semiLagrangian method. To this effect, this type of approach reaches highorder accuracies in both space and time. The objective of employing the Taylor expansion method for time is a convenient procedure that manages integration in time and increases temporal accuracy. In another relevant study [11], Crouseilles et al. presented several methods to approximate the solution of characteristics curves, but we adopt a RK method to accurately tract the characteristics curves. So this study utilizes WENO procedures to achieve a uniform highorder precision.
The remainder of this paper is organized as follows. Section 2 reviews FV WENO reconstruction for scalar cases. Section 3 describes the SLFV scheme and offers WENO reconstructions for both scalar and hyperbolic conservation law systems. In Section 4, experimental results confirming the efficiency of the approaches are provided. Section 5 summarizes and concludes the study, and it offers potential future research direction.
2 Review of FV method for scalar conservation laws
For highorder WENO reconstructions, say, of \(2k+1\)thorder of accuracy, we first need to recognize \(k+1\) small templates \(S_{\tau}\), \(\tau=0,\ldots,k\), the cell \(I_{j}\) is part of each of them. Generality, here we make \(S_{\tau}=\bigcup_{l=0}^{k} I_{j+\taul}\). We then utilize \(S= \bigcup_{\tau=0}^{k} S_{\tau}\) to describe the larger template, which consists of all the stencils from the \(k+1\) smaller templates. We built a kth degree polynomial reconstruction in each of the cells \(S_{\tau}\), \(\tau=0,\ldots,k\), which is indicated by \(H_{\tau}(x)\). Namely, the cell average of \(H_{\tau}(x)\) in the template \(S_{\tau}\) is in accord with the given cell average of v. Analogously, associated with the larger stencil S, we also find a \((2k)\)thorder polynomial, \(G(x)\). The details of reconstructing the polynomials \(H_{\tau}(x)\) and \(G(x)\) can be learned, for example, from [26].
The WENO algorithm is now well defined.
3 Highorder semiLagrangian FV method for onedimensional problems
In the part, the detailed procedures of the SLFV WENO 3 and WENO 5 approaches are designed for scalar and Euler equations.
3.1 SLFV method for the scalar problem
3.2 WENO reconstructions for the SLFV scheme of the scalar conservation laws
As everyone knows, the WENO schemes can achieve highorder precision and capture shock at the discontinuities. Next, we shall present the WENO reconstructions of third order and fifth order for the SLFV scheme, respectively.
3.2.1 \(g'(x,t)\geq{0}\) in the SLFV scheme
Thirdorder WENO reconstruction
We remark that when the CFL number is large enough to make \(x_{j+\frac {1}{2}}x_{0}^{(n)}>\Delta{x_{j}}\), the point \(x_{0}^{(n)}\) would be located at the corresponding meshes. So we build the WENO reconstruction to the corresponding template around \(x_{0}^{(n)}\).
Fifthorder WENO reconstruction
3.2.2 \(g'(x,t)<0\) in the SLFV scheme
Thirdorder WENO reconstruction
Fifthorder WENO reconstruction
The identical arithmetic is used to build this condition, so, simply, we use the templates \(S_{\tau}=\bigcup_{l=0}^{2} I_{j+1+\taul}\), \(\tau=0,1,2\), and \(S= \bigcup_{\tau=0}^{2} S_{\tau}\) to establish the fifthorder scheme in this case.
3.3 SLFV WENO schemes for the Euler equations

At the point \(x_{j+\frac{1}{2}}\), compute the average value for given cell averages \(\mathbf{v}_{j}\).$$ {\tilde{\mathbf{v}}_{j+\frac{1}{2}}}=\frac{1}{2}(\mathbf{v}_{j}+ \mathbf{v}_{j+1}). $$(3.6)

Compute the Jacobian matrix \(\mathbf{g}'({\tilde{\mathbf {v}}_{j+\frac{1}{2}}})\) denoted by \(\mathbf{A}_{j+\frac{1}{2}}\), the \(m\times m\) matrix \(\mathbf{R}_{j+\frac{1}{2}}({\tilde{\mathbf {v}}_{j+\frac{1}{2}}})\) composed of its right eigenvectors, and the \(m\times m\) matrix \(\mathbf{R}_{j+\frac{1}{2}}^{1}({\tilde {\mathbf{v}}_{j+\frac{1}{2}}})\) composed of its left eigenvectors$$\mathbf{A}_{j+\frac{1}{2}}=\mathbf{g}'({\tilde{ \mathbf{v}}_{j+\frac{1}{2}}}), \qquad \mathbf{R}_{j+\frac{1}{2}}= \mathbf{R}_{j+\frac{1}{2}}({\tilde {\mathbf{v}}_{j+\frac{1}{2}}}), \qquad \mathbf{R}_{j+\frac{1}{2}}^{1}=\mathbf {R}_{j+\frac{1}{2}}^{1}({ \tilde{\mathbf{v}}_{j+\frac{1}{2}}}). $$

For each target cell \(I_{j}\), project \(\mathbf{v}_{j}\) to the characteristic field locally by using the transformationwhere \(i = j2,\ldots, j+3\) for the fifthorder WENO reconstructions.$$\mathbf{u}_{i}=\mathbf{R}_{j+\frac{1}{2}}^{1} \mathbf{v}_{i}, $$

Perform scalar reconstructions on each component of u to get \(\mathbf{u} _{j+\frac{1}{2}}^{\pm}\).

\(\mathbf{v}_{j+\frac{1}{2}}^{\pm} = \mathbf{R}_{j+\frac{1}{2}}\mathbf {u}_{j+\frac{1}{2}}^{\pm}\).
Remark 3.1
Note that when the linear weights \(c_{0}\), \(c_{1}\), \(c_{2}\), and \(c_{3}\) are negative, the approach will become one of instability. Shi et al. [28] propose a method, which is used to tackle the negative weights, which involves few cost and is quite effective. The method can keep the scheme with negative weights steady and achieve highorder precision.
4 Numerical results
In this article, the mathematical simulations of thirdorder SLFV and fifthorder SLFV processes are provided for scalars and Euler equations. In our experiments, the CFL condition is bound to be 1.9 in the linear scalar equations, 2.9 in the nonlinear scalar conservation law, and 9.9 in the Euler systems. Uniform meshes are used in all simulations. The boundary conditions of all the examples are periodic in this paper.
4.1 The scalar problems
Example 1
(The linear conservation law)
The semiLagrangian finite volume WENO 3 and WENO 5 methods on Example 1 with \(\pmb{t=5}\) , \(\pmb{\mathrm{CFL}=1.9}\)
N  SLFV 3  SLFV 5  

\(\boldsymbol{L^{1}}\) error  Order  \(\boldsymbol{L^{\infty}}\) error  Order  \(\boldsymbol{L^{1}}\) error  Order  \(\boldsymbol{L^{\infty}}\) error  Order  
20  9.04E − 04    1.41E − 03    1.58E − 04    2.46E − 04   
40  1.41E − 04  2.68  2.22E − 04  2.67  5.05E − 06  4.97  7.92E − 06  4.96 
80  1.64E − 05  3.10  2.58E − 05  3.11  1.58E − 07  5.00  2.49E − 07  4.99 
160  2.09E − 06  2.97  3.29E − 06  2.97  5.31E − 09  4.90  8.35E − 09  4.90 
320  2.62E − 07  2.99  4.11E − 07  3.00  1.63E − 10  5.03  2.05E − 10  5.06 
Example 2
(The nonlinear conservation law)
The semiLagrangian finite volume WENO 3 and WENO 5 methods on Example 2 with \(\pmb{t=0.3}\) , \(\pmb{\mathrm{CFL}=2.9}\)
N  SLFV 3  SLFV 5  

\(\boldsymbol{L^{1}}\) error  Order  \(\boldsymbol{L^{\infty}}\) error  Order  \(\boldsymbol{L^{1}}\) error  Order  \(\boldsymbol{L^{\infty}}\) error  Order  
20  1.07E − 03    3.40E − 03    1.15E − 04    5.22E − 04   
40  1.30E − 04  3.04  4.47E − 04  2.93  4.05E − 06  4.83  2.08E − 05  4.65 
80  1.60E − 05  3.02  5.84E − 05  2.94  1.27E − 07  5.00  7.21E − 07  4.85 
160  1.98E − 06  3.01  7.19E − 06  3.02  3.96E − 09  5.00  2.31E − 08  4.96 
200  1.01E − 06  3.02  3.67E − 06  3.01  1.30E − 09  4.99  7.58E − 09  4.99 
4.2 Euler systems
Example 3
The semiLagrangian finite volume WENO 3 and WENO 5 methods on Example 3 with \(\pmb{t=2}\) , \(\pmb{\mathrm{CFL}=9.9}\)
N  SLFV 3  SLFV 5  

\(\boldsymbol{L^{1}}\) error  Order  \(\boldsymbol{L^{\infty}}\) error  Order  \(\boldsymbol{L^{1}}\) error  Order  \(\boldsymbol{L^{\infty}}\) error  Order  
20  1.91E − 03    2.99E − 03    4.01E − 05    6.22E − 05   
40  2.54E − 04  2.91  3.98E − 04  2.91  1.27E − 06  4.98  1.99E − 06  4.97 
80  3.22E − 05  2.98  5.05E − 05  2.98  4.10E − 08  4.95  6.15E − 08  5.02 
160  4.03E − 06  3.00  6.33E − 06  3.00  1.29E − 09  4.99  1.84E − 09  5.06 
200  2.07E − 06  2.99  3.24E − 06  3.00  4.24E − 10  4.99  5.73E − 10  5.23 
Example 4
(The Sod problem)
Example 5
(The Lax problem)
Example 6
(The stationary contact discontinuity problem)
Example 7
(The tiny density and internal energy problem)
Example 8
(The shock density wave interaction problem)
Example 9
(The interacting blast waves problem)
5 Concluding remarks
The highorder SLFV approaches presented above simulate the results of the scalar and Euler equations in this paper. The designed method, without restriction about the CFL time step, maintains uniform highorder precision in both space and time. Many onedimensional experiments are performed based on SLFV WENO 3 and WENO 5 reconstructions in order to validate the performance and efficiency of the proposed methods for scalar and conservation law systems. Further details regarding the construction of highorder SLFV methods for multidimensional equations are still under dispute, and these are the primary direction of future research.
Declarations
Acknowledgements
This work is supported by the National Science Foundation of China (11271100, 11301113, 71303067), Harbin Science and Technology Innovative Talents Project of Special Fund (2013RFXYJ044), China Postdoctoral Science Foundation funded project (Grant No. 2013M541400), the Heilongjiang Postdoctoral Fund (Grant No. LBHZ12102), the Fundamental Research Funds for the Central Universities (Grant No. HIT.HSS.201201).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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