Here, we mainly consider the 5th-order finite volume version because of its popularity and other cases can be obtained similarly. Consider the linear advection equation

u t ( x , t ) + c u x ( x , t ) = 0 , a ≤ x ≤ b , t ≥ 0. (7)

In this case, the flux function f ( u ( x , t ) ) = c u ( x , t ) . For this equation, the initial condition u 0 ( x ) simply propagates right if c > 0 (or left if c < 0 ) with unchanged shape.

For evaluating the average flux f i + 1 / 2 n in (4), we can firstly apply a 5th-order reconstruction based on piecewise constant average fluxes { f k } k = 1 N to obtain interpolation polynomial P i ( x ) on cell I i

f ( u ( x , t n ) ) = P i ( x ) + O ( Δ x 5 ) . (8)

The 5th-order polynomial P i ( x ) is reconstructed over the stencil

S i = { I i − 2 , I i − 1 , I i , I i + 1 , I i + 2 } .

Here, we assume the advection velocity c > 0 , then u ( x , t n ) simply spreads to the right with velocity c , which gives

f ( u ( x i + 1 / 2 , t ) ) = f ( u ( x i + 1 / 2 − c ( t − t n ) , t n ) ) (9)

for any time t ∈ [ t n , t n + 1 ] . Combining the Formulas (8) and (9), the flow rate f ( u ( x , t ) ) at cell edge x i + 1 / 2 is

f ( u ( x i + 1 / 2 , t ) ) = P i ( x i + 1 / 2 − c ( t − t n ) ) + O ( Δ x 5 ) . (10)

In this case, the average flux f i + 1 / 2 n can be expressed approximately as

f i + 1 / 2 n = 1 Δ t ∫ t n t n + 1 f ( u ( x i + 1 / 2 , t ) ) d t = 1 Δ t ∫ t n t n + 1 P i ( x i + 1 / 2 − c ( t − t n ) ) d t + O ( Δ x 5 ) = 1 c Δ t ∫ x i + 1 / 2 − c Δ t x i + 1 / 2 P i ( x ) d x + O ( Δ x 5 ) . (11)

Omitting the high-order term O ( Δ x 5 ) , we obtain the numerical flux

F i + 1 / 2 n = 1 c Δ t ∫ x i + 1 / 2 − c Δ t x i + 1 / 2 P i ( x ) d x . (12)

The last equality in (11) is obtained by integration of substitution x = x i + 1 / 2 − c ( t − t n ) . From the Equation (11), the integration in time [ t n , t n + 1 ] is transformed into integration in space [ x i + 1 / 2 − c Δ t , x i + 1 / 2 ] . Due to the integrand P i ( x ) is reconstruction polynomial, the last integration in (12) can be computed exactly.

Remark 1. This transformation is the main idea of this paper. And it is similar to the equation (3.9) in [16] but with a slightly different way. The main difference is that the operation in [16] transforms the flux integration in time at cell center into the integration of u ( x , t n ) in space

∫ t n t n + 1 f ( u ( x i , t ) ) d t = ∫ x i ∗ x i u ( x , t n ) d x ,

where x i ∗ is the backward characteristic point of cell center x i . And here we transform the flux integration in time at cell edge into the integration of interpolation polynomial of flux function in space. This difference makes the extension of the scheme to nonlinear cases much easier.

In the following of this subsection, we will present the 5th-order WENO reconstruction process to approximate the integral in (12)

∫ x i + 1 / 2 − c Δ t x i + 1 / 2 P i ( x ) d x .

The 5th-order WENO reconstruction procedure is represented as the convex combination of three 3rd-order reconstructions. First, we intend to reconstruct the 3rd-order conservative polynomials on cell I i = [ x i − 1 / 2 , x i + 1 / 2 ] based on the piecewise constant average fluxes.

{ f ¯ k n } , k = i − 2 , ⋯ , i + 2.

It is noted that there are 3 three-point stencils contained the cell I i can be used to reconstruct the polynomials. We denote these stencils respectively by

S i 0 = { I i , I i + 1 , I i + 2 } ,

S i 1 = { I i − 1 , I i , I i + 1 } ,

S i 2 = { I i − 2 , I i − 1 , I i } .

The corresponding polynomial of each stencil is

P i j ( x ) = b 0 j + b 1 j x + b 2 j x 2 , j = 0 , 1 , 2 , (13)

where the subscript i denotes the polynomial on cell I i and superscript j denotes the reconstruction based on stencil S i j . The coefficients in (13) are given by (for simplicity, we omit superscript in f i n )

b 0 j = 11 f ¯ i − j − 7 f ¯ i + 1 − j + 2 f ¯ i + 2 − j 6 + ( 2 f ¯ i − j − 3 f ¯ i + 1 − j + f ¯ i + 2 − j ) x i − 1 / 2 − j Δ x + ( f ¯ i − j − 2 f ¯ i + 1 − j + f ¯ i + 2 − j ) x i − 1 / 2 − j 2 2 Δ x 2 , b 1 j = − 2 f ¯ i − j − 3 f ¯ i + 1 − j + f ¯ i + 2 − j Δ x − ( f ¯ i − j − 2 f ¯ i + 1 − j + f ¯ i + 2 − j ) x i − 1 / 2 − j Δ x 2 , b 2 j = f ¯ i − j − 2 f ¯ i + 1 − j + f ¯ i + 2 − j 2 Δ x 2 ,

where j = 0 , 1 , 2 and Δ x = x i + 1 / 2 − x i − 1 / 2 is the length of cell I i . So far, we have obtained the conservative interpolation polynomials P i j ( x ) on each stencil S i j . In the end, the integral in (12) can be expressed as

∫ x i + 1 / 2 − c Δ t x i + 1 / 2 P i ( x ) d x = ∑ j = 0 2 d j ∫ x i + 1 / 2 − c Δ t x i + 1 / 2 P i j ( x ) d x ,

where the linear optimal weights d j are

d 0 = 3 10 − 1 4 λ + 1 20 λ 2 ,

d 1 = 6 10 + 1 10 λ − 1 10 λ 2 ,

d 2 = 1 10 + 3 20 λ + 1 20 λ 2 ,

where λ = c Δ t Δ x . For alleviating the effect of the non-smooth stencils, the nonlinear weights can be constructed as follows

w i j = a i j a i 0 + a i 1 + a i 2 (14)

and

a i j = d j ( ε + β i j ) 2 . (15)

The β i j is the indicator of smoothness based on the stencil S i j ,

β i 0 = 13 12 ( f ¯ i − 2 f ¯ i + 1 + f ¯ i + 2 ) 2 + 1 4 ( 3 f ¯ i − 4 f ¯ i + 1 + f ¯ i + 2 ) 2 ,

β i 1 = 13 12 ( f ¯ i − 1 − 2 f ¯ i + f ¯ i + 1 ) 2 + 1 4 ( f ¯ i − 1 − f ¯ i + 1 ) 2 ,

β i 2 = 13 12 ( f ¯ i − 2 − 2 f ¯ i − 1 + f ¯ i ) 2 + 1 4 ( f ¯ i − 2 − 4 f ¯ i − 1 + 3 f ¯ i ) 2 .

Finally, the numerical flux should be expressed as

F i + 1 / 2 n = 1 c Δ t ∑ j = 0 2 w i j ∫ x i + 1 / 2 − c Δ t x i + 1 / 2 P i j ( x ) d x . (16)

Substituting the Formulas (13) into (16) gives

F i + 1 / 2 n = 1 c Δ t ∑ j = 0 2 w i j ∫ x i + 1 / 2 − c Δ t x i + 1 / 2 P i j ( x ) d x = 1 6 [ w i 0 ( λ 2 ( f ¯ i − 2 f ¯ i + 1 + f ¯ i + 2 ) + λ ( 3 f ¯ i − 3 f ¯ i + 1 ) + ( 2 f ¯ i + 5 f ¯ i + 1 − f ¯ i + 2 ) ) + w i 1 ( λ 2 ( f ¯ i − 1 − 2 f ¯ i + f ¯ i + 1 ) + λ ( 3 f ¯ i − 3 f ¯ i + 1 ) + ( − f ¯ i − 1 + 5 f ¯ i + 2 f ¯ i + 1 ) ) + w i 2 ( λ 2 ( f ¯ i − 2 − 2 f ¯ i − 1 + f ¯ i ) + λ ( − 3 f ¯ i − 2 + 9 f ¯ i − 1 − 6 f ¯ i ) + ( 2 f ¯ i − 2 − 7 f ¯ i − 1 + 11 f ¯ i ) ) ] . (17)

In contrast, when the advection velocity c < 0 , u ( x , t n ) spreads to the left with velocity c , which gives

f ( u ( x i + 1 / 2 , t ) ) = f ( u ( x i + 1 / 2 − c ( t − t n ) , t n ) ) (18)

and the flow rate f ( u ( x , t ) ) at cell edge x i + 1 / 2 is

f ( u ( x i + 1 / 2 , t ) ) = P i + 1 ( x i + 1 / 2 − c ( t − t n ) ) + O ( Δ x 5 ) . (19)

Similarly, the average flux F i + 1 / 2 n can be expressed as

F i + 1 / 2 n = 1 − c Δ t ∑ j = 0 2 w i + 1 j ∫ x i + 1 / 2 x i + 1 / 2 − c Δ t P i + 1 j ( x ) d x . (20)

The nonlinear weight w i + 1 j can be constructed similarly as (14) and (15). The linear optimal weights are

d 0 = 1 10 − 3 20 λ + 1 20 λ 2 ,

d 1 = 6 10 − 1 10 λ − 1 10 λ 2 ,

d 2 = 3 10 + 1 4 λ + 1 20 λ 2 .

Substituting the formula (13) into (20) gives

F i + 1 / 2 n = 1 6 [ w i + 1 0 ( λ 2 ( f ¯ i + 1 − 2 f ¯ i + 2 + f ¯ i + 3 ) + λ ( 6 f ¯ i + 1 − 9 f ¯ i + 2 + 3 f ¯ i + 3 ) + ( 11 f ¯ i + 1 − 7 f ¯ i + 2 + 2 f ¯ i + 3 ) ) + w i + 1 1 ( λ 2 ( f ¯ i − 2 f ¯ i + 1 + f ¯ i + 2 ) + λ ( 3 f ¯ i − 3 f ¯ i + 1 ) + ( 2 f ¯ i + 5 f ¯ i + 1 − f ¯ i + 2 ) ) + w i + 1 2 ( λ 2 ( f ¯ i − 1 − 2 f ¯ i + f ¯ i + 1 ) + λ ( 3 f ¯ i − 3 f ¯ i + 1 ) + ( − f ¯ i − 1 + 5 f ¯ i + 2 f ¯ i + 1 ) ) ] . (21)

The scheme proposed in the paper is conservative for linear and nonlinear cases. The property of conservation is obvious since the scheme (6) is represented in conservation form.

Remark 2. In [16] , the authors Qiu and Shu presented a semi-Lagrangian finite difference WENO method for advection in incompressible flow. Actually, there are two reconstruction procedures needed to obtain the numerical flux at cell edge x i + 1 / 2 . It is noted that, for linear equation with constant coefficient, the formulas of average flux (17) and (21) are totally same as ones in [16] . However, these two methods are devised by totally different procedure. They mainly have the following differences: the method present here is finite volume; our method can be easily extended to nonlinear cases by freezing locally and detailed description is given in Subsection 2.3; there are always exist linear optimal weights to construct the WENO schemes for our scheme which is not satisfied the schemes in [16] .

Here, we conclude the algorithm for computing numerical flux F i + 1 / 2 n at x = x i + 1 / 2 during a time step Δ t = t n + 1 − t n .

1) Distinguish the propagation direction of solution at x = x i + 1 / 2 used Rankine-Hugoniot jump condition

c i + 1 / 2 = f ( u i + 1 ) − f ( u i ) u i + 1 − u i .

2) The propagation velocity can be chosen to be c = ∂ f ( u ) ∂ u | u = u i if c i + 1 / 2 > 0 , or c = ∂ f ( u ) ∂ u | u = u i + 1 if c i + 1 / 2 < 0 .

3) Insert the propagation velocity into (17) or (21) to get numerical flux F i + 1 / 2 n .

In this subsection, we will extent the method above to nonlinear case. Firstly, it is noted that for nonlinear case the method here cannot approach the optimal accuracy since the solution no longer simply translates uniformly. Instead, it deforms as it evolves and the characteristic curves are no longer parallel straight lines, which lead to the evaluation of average fluxes is hard to obtain. 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 equation to linear formation locally and apply the procedure in Subsection 2.1 to it. For solving the nonlinear case, the propagation direction is firstly distinguished by Rankine-Hugoniot jump

conditions and propagation velocity is chosen to be c = ∂ f ( u ) ∂ u | u = u i ( ifpropagationdirection > 0 ) or c = ∂ f ( u ) ∂ u | u = u i + 1 ( ifpropagationdirection < 0 ) .

Remark 3. The semi-Lagrangian finite volume method proposed here is largely dependent on the reconstruction from the cell flux average

f ¯ i = 1 Δ x ∫ x i − 1 / 2 x i + 1 / 2 f ( u ( x , t n ) ) d x .

For finite volume method, only the cell average

u ¯ i n = 1 Δ x ∫ x i − 1 / 2 x i + 1 / 2 u ( x , t n ) d x

is available. For linear problem, f ( u ( x , t ) ) = a u ( x , t ) , the cell flux average can be apparently chosen to be

f ¯ i = a u ¯ i n

However, for nonlinear problem, the cell flux average must be computed by some numerical quadrature. In this paper, we choose Gauss quadrature to evaluate the cell flux average

f ¯ i = 1 Δ x ∫ x i − 1 / 2 x i + 1 / 2 f ( u ( x , t n ) ) d x = 1 Δ x ∑ k = 1 k = 3 A k g f ( u ( x k g , t n ) ) ,

where A k g and x k g are the Gauss weights and Gauss points, respectively. For simple, the value of u ( x , t n ) at Gauss point x k g can be obtained by the 5th- order Lagrangian interpolation.

In addition, for solving Euler equations, the Roe speed is used to identify the upwind direction which may lead to some numerical instabilities, especially for 2D cases. In Subsection 3.3, in computing the numerical tests: double Mach reflection and Mach 3 wind tunnel with a step, there indeed exist numerical instabilities. In addition, it is noted that this numerical instability not only appears in the WENOEL scheme, but also appears in the WENOJS scheme. It is said that the numerical instability is mainly due to the choice of Roe speed. The detailed introductions for the numerical instabilities of upwind schemes are referred to [20] [21] [22] . In order to eliminate such kind of instabilities, the H-correction procedure in [21] is taken to solve these flaws. For these two 2D problems, the numerical flux F i + 1 / 2 , j n at the cell edge ( x i + 1 / 2 , y j ) is computed as follows. [1]

1) If min ( | λ i , j | , | λ i + 1 , j | ) ≥ η i + 1 / 2 , j , then the numerical flux F i + 1 / 2 , j n can be chosen to be (17) or (21) and the average fluxes f k in these two equations are correspondingly substituted by f k , j ( k = i − 2 , ⋯ , i + 3 ) . The λ i , j is the eigenvalue and the value η i + 1 / 2 , j is determined by

η i + 1 / 2 , j = max ( η i + 1 / 2 , j x , η i , j − 1 / 2 y , η i , j + 1 / 2 y , η i + 1 , j − 1 / 2 y , η i + 1 , j + 1 / 2 y ) ,

where

η i + 1 / 2 , j x = | u i , j − u i + 1 , j | + | a i , j − a i + 1 , j | ,

η i , j + 1 / 2 y = | v i , j − v i , j + 1 | + | a i , j − a i , j + 1 | ,

and a i , j is the speed of sound.

2) Otherwise, a more dissipative Lax-Friedrich flux splitting method is used to split the flux f k , j into positive and negative fluxes

f k , j + = 1 2 ( f k , j + α u k , j n ) ,

f k , j − = 1 2 ( f k , j − α u k , j n ) .

For the positive and negative fluxes, the propagation velocities are chosen to be

c + = 1 2 ( λ k , j + α ) ,

c − = 1 2 ( λ k , j − α ) .

Inserting the f k , j + , c + and f k , j − , c − into (17) and (21), respectively, we can get the positive and negative numerical fluxes F i + 1 / 2 , j + and F i + 1 / 2 , j − . Finally, the numerical flux at cell edge ( x i + 1 / 2 , y j ) is

F i + 1 / 2 , j n = F i + 1 / 2 , j + + F i + 1 / 2 , j − .

In this subsection, we will consider linear advection, Burger’s and Buckley-Le- verett equations. For the linear advection equation, two initial conditions are used to test the schemes. We use a smooth initial condition to test the order of accuracy of the scheme and a condition contained discontinuities and high-fre- quency waves to test the performance of the scheme in simulating discontinuous and large-gradient solution. For the Burger equation, we also consider two initial conditions which commonly used to validate the proposed scheme. For the nonconvex Buckley-Leverett equation, we study a simple model for two-phase fluid flow in a porous medium. Except for the comparison of resolution between the WENOJS and WENOEL schemes, the computing time is also shown in this subsection.

In this subsection, we consider 1D Euler equations since one of the main application areas of high-resolution scheme is compressible gas dynamics,

( ρ ρ u E ) t + ( ρ u ρ u 2 + p ( E + p ) u ) x = 0 , (30)

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:

E = p γ − 1 + 1 2 ρ u 2 ,

where the ratio of specific heats γ = 1.4 .

The following three initial conditions combined with Euler Equations (30) are considered, which are often used to examine the methods for solving Euler equations:

( ρ , u , p ) = { ( 0.445 , 0.698 , 3.528 ) , if − 5 ≤ x < 0 , ( 0.5 , 0 , 0.571 ) , if 0 ≤ x ≤ 5 , (31)

( ρ , v , p ) = { ( 27 7 , 4 35 9 , 31 3 ) , if − 5 ≤ x < − 4 , ( 1 + 0.2 sin ( 5 x ) , 0 , 1 ) , if − 4 ≤ x ≤ 5. (32)

( ρ , u , p ) = { ( 1 , 0 , 1000 ) , if − 5 ≤ x < − 4 , ( 1 , 0 , 0.01 ) , if − 4 ≤ x < 4 , ( 1 , 0 , 100 ) , if 4 ≤ x ≤ 5 , (33)

For solving Euler equations, Roe linearization (i.e. Roe average) is used to locally freeze a nonlinear system to a linear system. Then this system is de- coupled into three advection equations and each equation is solved by procedure in Section 0. It is noted that the propagation direction of solution of each advection equation is distinguished by corresponding eigenvalue of Roe matrix.

For the problem (30) (31), called Lax problem [26] , we solve it with Δ t = 0.6 Δ x / λ for the WENOEL and WENOJS methods. Figure 6 is computed with cells N = 200 and output time t = 1.3 . Apparently, the WENOEL method performs almost the same as WENOJS method and presents a robust solution for this shock-tube problem. The exact solution is obtained by the solver presented in [24] .

The problem (30) (32), called shock entropy wave interaction [4] , is usually used to test the high-order methods for capturing the high-frequency waves. The CFL number is also set to be Δ t = 0.6 Δ x / λ for the WENOEL and WENOJS methods. We evolve the equations up to t = 1.8 with cells N = 400 . For this problem, a Mach 3 shock wave moved right interacts with sine waves in a density which lead to a field obtained smooth and discontinuous structures. The reference solution in Figure 7 is computed by WENOJS method with cells N = 2000 . It is plainly seen from Figure 7 that, as WENOJS, the WENOEL method performs with the high resolution. Furthermore, from Figure 8, we can find that in the part of high-frequency waves the WENOEL method performs less dissipative than the WENOJS method.

The problem (30) (33) was originally proposed as a benchmark for testing several numerical methods by Woodward and Colella [19] . The reflective boundary conditions are applied to both boundaries. The CFL number is also set to be Δ t = 0.6 Δ x / λ for the WENOEL and WENOJS methods. We evolve the equations up to t = 0.38 with cells N = 400 . The reference solution in Figure 9 is computed by WENOJS method with cells N = 2000 . As the last two problems, the WENOEL method gives almost the same resolution as WENOJS method.

Finally, we consider a numerical experiment for 2D Euler equations for gas dynamics,

[ ρ ρ u ρ v E ] t + [ ρ u ρ u 2 + p ρ u v ( E + p ) u ] x + [ ρ v ρ u v ρ v 2 + p ( E + p ) v ] y = 0, (34)

where the equation of state is

p = ( γ − 1 ) ( E − 1 2 ρ ( u 2 + v 2 ) ) , γ = 1.4.