The Approximated Semi-Lagrangian WENO Methods Based on Flux Vector Splitting for Hyperbolic Conservation Laws

The paper is devised to combine the approximated semi-Lagrange weighted essentially non-oscillatory scheme and flux vector splitting. The approximated finite volume semi-Lagrange that is weighted essentially non-oscillatory scheme with Roe flux had been proposed. The methods using Roe speed to construct the flux probably generates entropy-violating solutions. More seriously, the methods maybe perform numerical instability in two-dimensional cases. A robust and simply remedy is to use a global flux splitting to substitute Roe flux. The combination is tested by several numerical examples. In addition, the comparisons of computing time and resolution between the classical weighted essentially non-oscillatory scheme (WENOJS-LF) and the semi-Lagrange weighted essentially non-oscillatory scheme (WENOEL-LF) which is presented (both combining with the flux vector splitting).


Introduction
The semi-Lagrangian methods are popularly used in weather prediction [1] [2] and simulation of the Vlasov equations [3] [4] [5] [6], and so on.These methods solve the problems with a characteristic tracing algorithm.It is this property of algorithm that leads to the following two advantages: no time discretization and the alleviation of CFL time step restriction.The semi-Lagrangian methods (combined wih WENO reconstruction [7] [8]) used to solve hyperbolic problems are presented in [9] [10] [11].
In [10], the authors proposed a finite volume semi-Lagrangian WENO scheme for advection problems.The scheme combined the Eulerian-Lagrangian F. X. Hu framework [12] [13] with high-order WENO reconstruction, and used a integralbased WENO reconstruction to handle trace-back integration.In this framework, the scheme trace each computational Eulerian grid cell at time 1 n t + backward over a time step along the characteristic line to its Lagrangian track-back region at n t .The average mass is simply transported from the Lagrangian region to the computational grid cell.In addition, they presented a theoretical proof for the accuracy of the method.
In [11], the authors developed a finite volume semi-Lagrangian WENO scheme for nonlinear conservation laws.This method can be regarded as the extension of the method for advection problems in [10].A problem appeared in such extension is that one does not have the particle velocity since it is nonlinearly related to the unknown solution.Hence, one cannot find the exact tracking line of fluid particle.Instead of trying to find the the exact characteristic line of particle, they use a known velocity for the tracing line computation.Since this will not give the correct tracing line, a flux correction procedure is needed to increase the accuracy and numerical stability.The proposed method arrives optimal order of accuracy.However, the procedure of flux correction makes the scheme rather cost to be implemented.
In [9], a sort of finite difference semi-Lagrangian ENO and WENO schemes are devised for advection problems in incompressible flow.The key part of this paper is that the integral form of advection equation is taken over a triangle region.This integral procedure transforms the flux integration in time at cell center into the integration of mass ( ) where i x * is backward characteristic point of cell center i x .This scheme is difficult to extent to nonlinear cases.In addition, there is no optimal linear weights for WENO schemes for solving advection equations with variable coefficients (or also for nonlinear cases).
In [14], the authors developed an approximated finite volume semi-Lagrangian WENO schemes (WENOEL-Roe) for 1D and 2D nonlinear hyperbolic prolems.The scheme integrates the hyperbolic equation over the control volume  to obtain a integral equation.They try to directly evaluate the integration of flux function in time at cell edge For linear cases, the integration of flux function in time can be transformed into the integration of interpolation polynomial of flux average in space.For nonlinear cases, a local freeze method is used to freeze the nonlinear into linear cases.The scheme here is different with the traditional semi-Lagrangian scheme [10] [11].The backward tracing characteristic point is needed in the procedure of evaluating the integration of flux function.The advantages of the WENOEL-Roe scheme are easy implement and high efficiency.Refer to [14] for a detail.
The procedure of evaluating the integration of flux function also depends on the direction of upwind.The Roe average speed is used to identified the upwinding.It is known that the Roe schemes maybe generates entropy-violating solutions.More seriously, these scheme can perform numerical instability for some two-dimensional problems [15] [16] [17].A local entropy correction can be used to remedy this deficiency.However, it is usually more robust to use a global flux splitting.In this paper, an approximated finite volume semi-Lagrangian WENO scheme with the smooth Lax-Friedrichs flux splitting (WENOEL-LF) is presented.The WENOEL-LF scheme is less resolution than WENOEL-Roe since the Lax-Friedrichs flux splitting method is more dissipative than Roe method.However, the advantage of WENOEL-LF scheme is it generally generates the entropy solution.
In the paper that follows, we will review the WENOEL-Roe scheme briefly in Section 2. In Section 3, we will present the formulation of WENOEL-LF scheme in Section 3 in detail.The comparisons of resolution and computing time between the WENOEL-LF and WENOJS-LF schemes is presented in Section 4.

Review the WENOEL-Roe Scheme
Consider the linear advection equation ( ) Denote the i -th cell average by ( ) and average flux at cell edge then (2) can be written as ( ) Denote the value n i U approximates the average value j i u and the numerical flux Then we obtain ( ) For evaluating average flux .
Here, we assume the advection velocity 0 a > , then ( ) simply spreads to the right with velocity a , which gives , , in any time when Combining the formulas ( 6) with ( 7), the flow rate , .
In this case, the average flux Omitting the high-order term ( ) , we obtain the numerical flux ( ) The last equality in ( 9) is obtained by integration of substitution ( ) . From the Equation ( 9), the integration in time [ ] Due to the integrand ( ) i P x is reconstruction polynomial, the last integration in (10) can be computed exactly.
In the following of this section, we will present the 5th-order WENO reconstruction process to approximate the integral in (10) ( ) The 5th-order WENO reconstruction procedure is represented as the convex combination of three 3rd-order reconstructions.First, we intend to reconstruct three 3rd-order conservative polynomials ( ) where the subscript i denotes the polynomial on cell i I and superscript j denotes the reconstruction based on stencil j i S .So far, we have obtained the conservative interpolation polynomials ( ) In the end, the integral in (10) can be expressed as ( ) ( ) where j d are the optimal weights.To alleviate the effect of the non-smooth stencils, the nonlinear weights can be constructed as follows where j i β is the indicator of smoothness of the polynomial on the stencil j i S .
Finally, the numerical flux should be expressed as ( ) Substituting the formula of polynomial ( ) In contrast, when the advection velocity 0 a < , ( ) spreads to the left with velocity a , which gives , , and the flow rate , .
Similarly, the average flux The nonlinear weight 1 j i w + can be constructed similarly as (11) and (12).
Substituting the formula of polynomial ( ) The above process of approximating average flux is reasonable for linear advection equation.For nonlinear problems, the formulas ( 9) and ( 17) is not hold any more since the solution no longer simply translates uniformly.And generally the tracking back points cannot be found exactly (even cannot find the points with high accuracy).Hence for nonlinear case, rather than trying to find the tracking back points; we freeze the nonlinear equation to linear formation locally and apply the procedure above to it.For solving the nonlinear case, the propagation direction is distinguished by Rankine-Hugoniot jump conditions and propagation velocity is chosen to be ( )

Formulation of WENOEL-LF Scheme
In this section, we solve nonlinear problems by using a more robust global flux splitting where ( ) ( ) Insert the Equation ( 19) into ( 1) and ( 2), we can obtain , which is similar to conservation formula (4), and denote A scheme approximated (20) can be written as This scheme is conservative.Since if we sum U + over the whole set of cells, we obtain .
+ denote the fluxes at the extreme edges.The sum of the flux differences cancels out except for the fluxes at the extreme.
The simplest smooth flux splitting we chose is the Lax-Friedrichs splitting where α is taken as ( ) over the whole set of cell averages.Since x + is chosen as ( ) . Similarly, The flow velocity a − for flux ( ) 14) and ( 18), respectively, the numerical fluxes can be obtained as follows, ))

Algorithm
Here, we conclude the algorithm for computing the approximated solution 4) Insert the numerical fluxes computed above into (21), we obtain the approximated solution

Numerical Results
In this section, we use several 1D and 2D nonlinear examples to test the WENOEL-LF scheme.The comparisons of resolution and computing time between the WENOEL-LF and WENOJS-LF is presented.It is found that, with the same number of cells, the WENOJS-LF scheme has slightly higher resolution than WENOEL-LF scheme.However, the computing time of WENOJS-LF scheme is almost two times for scalar equation (and almost three times for nonlinear system) over that of WENOEL-LF scheme.

Burgers' Equation
Consider the inviscid Burgers' equation with two initial conditions: ( ) and The Burger's Equation ( 24) with discontinuous initial condition (25) develops the solution which consists of a rarefaction wave and a shock wave.The numerical solutions computed by the WENOEL-LF and WENOJS-LF schemes are shown in Figure 1.These two solutions are both computed with 100 N = and CFL = 0.1.The final output time is chosen to be 1 t = .From Figure 1, we can find that the solution of WENOJS-LF scheme has slightly better resolution than that of WENOEL-LF scheme, especially around the rarefaction wave.And, in solving nonlinear cases (including the following tests), the WENOJS-LF scheme generally possesses slightly higher resolution than WENOEL-LF scheme when the same amount of cells is used.
Although the WENOJS-LF scheme has higher resolution than WENOEL-LF scheme, the WENOEL-LF scheme has the advantage of decreasing the computing time.Table 1 presents  that of the WENOJS-LF scheme.For comparing the efficiency between these two schemes, we plots the relationship between the 1 l error and computing time for these two schemes.From Figure 2, we can find that the efficiency of WENOEL-LF is higher than that of WENOJS-LF scheme.That is, to achieve the same 1 l error, the WENOEL-LF needs less computing time.
For the problem (24) (26), the initial condition is smooth, and the solution evolves discontinuity at 1 π t = .The Figure 3 is plotted with 100 N = and output time 0.3 t = . From this figure, no obviously difference is presented (however, by closed inspection, the resolution of WENOJS-LF is still slightly better than WENOEL-LF scheme).Similar to the last test, the superiority of our scheme lies in decreasing the computing time.Therefore, in efficiency, the WENOEL-LF scheme is still advantageous over the WENOJS-LF scheme.

The 1D Euler Equation
In this subsection, we consider 1D Euler equations since one of the main application areas of high-resolution scheme is compressible gas dynamics, ( ) where ρ , u , p , E are density, velocity, pressure and total energy, respectively.The system of equations is closed by the equation of state for an ideal polytropic gas: where the ratio of specific heats For the problem (27) (28), called Lax problem [18], we solve it with 0.1 t x λ ∆ = ∆ for the WENOEL-LF and WENOJS-LF methods.The Figure 4 shows the numerical solutions computed by the WENOEL-LF and WENOJS-LF schemes with 200 N = . From this figure, no obviously difference can be found between these two schemes and they both present high-resolution results.To make further comparison, we show the 1 l error and computing time in Table 2.
The exact solution of Riemann problem (27) (28) is generated by the code of E.
F. Toro in [19].From this table, we can find that, with the same number of cells,  For solving the problem (27) (29), we have the same set as the previous example but with different number of cells.Because no exact solution can be obtained, there is no comparison of error.In Figure 6, we show the numerical  , respectively.The "×" denotes the solutions computed by the WENOEL-LF scheme with 600 N = ., respectively.The "×" denotes the solutions computed by the WENOEL-LF scheme with 600 N = .
Figure 9.The zoomed version of Figure 8 around complicated region.

The 2D Euler System
Finally, we consider a numerical experiment for 2D Euler equations for gas dynamics, ( ) ( ) where the equation of state is ( ) ( ) Here, we apply the WENOEL-LF and WENOJS-LF schemes to the 2D double-Mach shock reflection problem where a strong vertical shock moves horizontally into a wedge which inclined with some angle with the ratio of specific heats 1.4 γ = .Initially, this problem was proposed by Woodward and Colella [20] and had been taken extensively as a test example for high-order schemes.The computational domain is chosen to be [ ] [ ] 0, 4 0,1 × and the reflective wall lies on the bottom of the computational domain for In Figure 10, we plot the numerical solution of the WENOJS-LF scheme with cells 400 × 100 and the numerical solutions of the WENOEL-LF scheme with cells 400 × 100 and 560 × 140, respectively.As the conclusion of 1D cases, with the same number of cells, the WENOJS-LF scheme can obtain higher-resolution solutions than the WENOEL-LF scheme, but with much more computing time.
When we increase the number of cells from 400 × 100 to 560 × 140, the numerical solution of the WENOEL-LF scheme performs higher resolution but with almost the same computing time as the WENOJS-LF scheme with 400 × 100.

Conclusion
This paper aims to combine the finite volume semi-Lagrangian WENO method with flux vector splitting.The proposed scheme is rather robust and easily implemented.And the computing time of the WENOEL-LF scheme is about half and one third of the WENOJS-LF scheme in scalar equation and nonlinear system, respectively.But with the same number of cells, unlike the performance in [14], the numerical solution of WENOJS-LF scheme is slightly better than that of WENOEL-LF scheme.If we compare the 1 L error with computing time (or the resolution of numerical solution and computing time), then we can find the WENOEL-LF scheme possesses higher efficiency than the WENOJS-LF scheme.That is, to obtain the numerical solution with the same resolution, the WENOEL-LF scheme needs less computing time.
+ in (3), we can firstly apply a 5th-order reconstruction based on piecewise constant average fluxes { } 1 the cell edges.The flow velocity a + for flux ( )

1 ) 2 )
Split the flux function ( ) f u into the positive flux Determine the flow velocities a + and a − of the positive flux

Figure 2 .
Figure 2. The comparison of efficiency for problem (24) (25) is presented between the WENOEL-LF and WENOJS-LF methods.

Figure 5 .
Figure 5.The comparison of efficiency for problem (27) (28) is presented between the WENOEL-LF and WENOJS-LF methods.

Figure 6 .
Figure 6.The numerical solutions for problem (27) (29) is computed by the WENOEL-LF and WENOJS-LF methods.The "+" and "  " denotes the solutions computed by the WENOEL-LF and WENOJS-LF schemes with 400 N =, respectively.The "×" denotes the solutions computed by the WENOEL-LF scheme with 600 N = .

Figure 8 .
Figure 8.The numerical solutions for problem (27) (30) is computed by the WENOEL-LF and WENOJS-LF methods.The "+" and "  " denotes the solutions computed by the WENOEL-LF and WENOJS-LF schemes with 400 N =, respectively.The "×" denotes the solutions computed by the WENOEL-LF scheme with 600 N = .
the beginning, a Mach 10 shock, moving right, is located at 1 6 x = , 0 y = and makes an angle 60˚ with the x-axis.For the boundary conditions, the exact postshock condition is imposed for bottom boundary from0 x = to 1 6 x = ,and the reflective boundary condition is imposed for the rest; the flows are imposed on the top boundary such that there is no interaction with the Mach 10 shock; inflow and outflow boundary conditions are set for the left and right boundaries respectively.The unshocked fluid has a density of 1.4, a pressure of 1 and this problem is run until 0.2 t = .

Figure 10 .
Figure 10.Double Mach problem.(a) The density contour is computed by the WENOEL-LF scheme with 400 × 100; (b) The density contour is computed by the WENOEL-LF scheme with 560 × 140; (c) The density contour is computed by the WENOJS-LF scheme with 400 × 100.30 equally spaced contour lines are plotted from 1.731 to 22.9705.

Table 1 .
The comparisons of computing time (in seconds) and 1 l error for initial value problem (24) (25) between the WENOEL-LF and WENOJS-LF methods.

Table 2 .
The comparisons of computing time (in seconds) and 1 l error for the problem (27) (28) between the WENOEL-LF and WENOJS-LF methods.