Lecture 1: Finite Volume WENO Schemes Chi-Wang Shu

Lecture 1: Finite Volume WENO Schemes Chi-Wang Shu Division of Applied Mathematics Brown University

Outline of the First Lecture General description of finite volume schemes for conservation laws The WENO reconstruction procedure Bound-preserving limiter for high order finite volume WENO schemes A simple WENO limiter for discontinuous Galerkin methods Concluding remarks

Finite volume schemes for conservation laws We look first at the one-dimensional hyperbolic conservation law u t + f(u) x = 0 which has discontinuous solutions even if the initial condition is smooth. We discretize the computational domain into cells I i = [x i 1/2,x i+1/2 ] with cell sizes x i (not necessary to be uniform or smooth varying). The cell averages are denoted by ū i = 1 x i xi+1/2 x i 1/2 u(x)dx.

A finite volume scheme approximates this conservation law in its integral form d dtūi + 1 x i ( f(ui+1/2 ) f(u i 1/2 ) ) = 0 (1) To convert (1) to a finite volume scheme, we take our computational variables as the cell averages ū i, i = 1, 2,,N and use a reconstruction procedure to obtain an approximation to u i+1/2.

A typical reconstruction procedure is to choose several consecutive cells near x i+1/2, which typically include at least one of I i and I i+1, the two neighbors of x i+1/2. The collection of these cells are called the stencil of the reconstruction. We seek a polynomial p(x) (or another simple function such as a trigonometric or exponential function) whose cell average over each cell I j in the stencil agrees with the given cell average ū j. We then take u i+1/2 = p(x i+1/2 ).

In order to obey upwinding for stability, we replace f(u i+1/2 ) by ( ) ˆf u i+1/2,u+ i+1/2 where ˆf(u,u + ) is a monotone numerical flux satisfying 1. ˆf(u,u + ) is non-decreasing in its first argument u and non-increasing in its second argument u +, symbolically ˆf(, ); 2. ˆf(u,u + ) is consistent with the physical flux f(u), i.e. ˆf(u,u) = f(u); 3. ˆf(u,u + ) is Lipschitz continuous with respect to both arguments u and u +. Here, both u i+1/2 and u+ i+1/2 are obtained through the reconstruction procedure, with their stencil biased to the left and to the right, respectively.

Time discretization can be achieved by the TVD (also called SSP) Runge-Kutta or multi-step methods Shu and Osher JCP 1987; Gottlieb, Shu and Tadmor SIAM Rev 2001; Gottlieb, Ketcheson and Shu, World Scientific 2011. For example, the third order TVD Runge-Kutta scheme is ū (1) = ū n + tl(ū n ) ū (2) = 3 4ūn + 1 4ū(1) + 1 4 tl(ū(1) ) ū n+1 = 1 3ūn + 2 3ū(2) + 2 3 tl(ū(2) )

One advantage of finite volume schemes is that they can be generalized to multi-dimensions including unstructured meshes easily in principle, even though the reconstruction procedure and the computation of numerical fluxes become more complicated, especially for unstructured meshes.

The WENO reconstruction procedure We would like to have schemes which are both high order accurate in smooth regions, and (essentially) non-oscillatory with sharp shock transition For typical linear schemes, i.e. schemes which are linear for a linear PDE u t + au x = 0 (2) (this corresponds to the situation that the stencil relative to the point x i+1/2 is fixed, for example, it is always {I i 1,I i,i i+1 }), the two properties desired above cannot be fulfilled simultaneously (Godunov Theorem).

We would therefore need to consider nonlinear schemes, which are nonlinear even for the linear PDE (2), such as the WENO schemes. In fact, the nonlinearity of the algorithm is only at the stage of choosing stencils in the reconstruction. Essentially non-oscillatory (ENO) reconstruction (Harten, Engquist, Osher and Chakravarthy, JCP 1987): Uniform high order polynomial reconstruction; The stencil is locally adaptive: among several candidate stencils one is chosen according to local smoothness.

j-2 S1 j j+1/2 j+2 j-1 S0 j+1 S2 Figure 1: Three possible stencils for reconstructing the point value at x j+1/2 using three cells in each stencil.

Weighted ENO (WENO) reconstruction (Liu, Osher and Chan, JCP 1994; Jiang and Shu, JCP 1996): Instead of using just one candidate stencil; a linear combination of all candidate stencils is used The choice of the weight to each candidate stencil, which is a nonlinear function of the cell averages, is a key to the success of WENO.

Advantages of ENO and WENO schemes: Uniform high order accuracy in smooth regions including at smooth extrema, unlike second order TVD schemes which degenerate to first order accuracy at smooth extrema; Sharp and essentially non-oscillatory (to the eyes) shock transition; Robust for many physical systems with strong shocks; Especially suitable for simulating solutions containing both discontinuities and complicated smooth solution structure, such as shock interaction with vortices.

Some advantages of WENO schemes over ENO schemes: Higher order of accuracy with the same set of candidate stencils: the order of accuracy is 3 instead of 2 for piecewise linear, and 5 instead of 3 for piecewise quadratic; No logical if statements in the stencil choosing process of ENO. Cleaner programming; Numerical flux function is smoother: C instead of only Lipschitz as in the ENO case. Hence: (i) a convergence proof when the solution is smooth (Jiang and Shu, JCP 1996), and (ii) better steady state convergence (Zhang and Shu, JSC 2007; Zhang, Jiang and Shu, JSC 2011; Hao et al., JCP submitted).

The WENO reconstruction procedure Given the cell averages ū i = 1 x i xi+1/2 x i 1/2 u(x)dx of a piecewise smooth function u(x) for the cells I i = [x i 1/2,x i+1/2 ] with cell sizes x i, find an approximation to the function u(x) at a desired location, e.g. at the cell boundaries x i+1/2. General procedure of reconstruction with a given stencil. e.g. to reconstruct u i+1/2 given ū i 1, ū i and ū i+1 :

1. Find the unique second order polynomial p(x) which agrees with the three given cell averages ū i 1, ū i and ū i+1 for the three cells in the stencil, respectively: 1 x i 1 1 x i 1 x i+1 xi 1/2 x i 3/2 p(x)dx = ū i 1, xi+1/2 x i 1/2 p(x)dx = ū i, xi+3/2 x i+1/2 p(x)dx = ū i+1. 2. Take the value p(x i+1/2 ) as an approximation to u i+1/2 : u i+1/2 = p(x i+1/2 )

3. The approximation u i+1/2 can be written out eventually as a linear combination of the given cell averages ū i 1, ū i and ū i+1 because the procedure is linear: u i+1/2 = 1 6ūi 1 + 5 6ūi + 1 3ūi+1 This approximation is third order accurate if the function u(x) is smooth in the stencil {I i 1, I i, I i+1 }.

Using such approximations with a fixed stencil leads to high order linear schemes, which will be oscillatory in the presence of shocks by the Godunov Theorem. The general procedure of a WENO reconstruction: 1. Compute the approximations from several different substencils, e.g. the three stencils in Figure 2:

j-2 S1 j j+1/2 j+2 j-1 S0 j+1 S2 Figure 2: Three sub-stencils for the reconstruction at x j+1/2 using three cells in each stencil.

u (0) i+1/2 = 1 3ūi 2 7 6ūi 1 + 11 6 ūi u (1) i+1/2 = 1 6ūi 1 + 5 6ūi + 1 3ūi+1 u (2) i+1/2 = 1 3ūi + 5 6ūi+1 1 6ūi+2 If the function u(x) is smooth in all three substencils, then the three approximations u (0) i+1/2, u(1) i+1/2 and u(2) i+1/2 are all third order accurate.

2. Find the combination coefficients γ 0, γ 1 and γ 2, also called linear weights, such that the linear combination u i+1/2 = γ 0 u (0) i+1/2 + γ 1u (1) i+1/2 + γ 2u (2) i+1/2 is fifth order accurate if u(x) is smooth in all substencils. This can be easily achieved with γ 0 = 1 10, γ 1 = 3 5 and γ 2 = 3 10 : u i+1/2 = 1 10 u(0) i+1/2 + 3 5 u(1) i+1/2 + 3 10 u(2) i+1/2 would lead to a fifth order accurate linear scheme which is oscillatory. At this stage, if we only require the reconstructed value u i+1/2 to be of the same order of accuracy as that from each of the substencils (in this case, third order rather than fifth order), then we can choose the linear weights γ k > 0 arbitrarily as long as they sum to one.

3. Find the nonlinear weights w 0, w 1 and w 2 such that u i+1/2 = w 0 u (0) i+1/2 + w 1u (1) i+1/2 + w 2u (2) i+1/2 is both fifth order accurate in smooth regions and non-oscillatory for shocks. Thus we require the nonlinear weights w 0, w 1 and w 2 to satisfy the following two properties: If u(x) is smooth in all three substencils, then the nonlinear weights w 0, w 1 and w 2 are close to the linear weights γ 0, γ 1 and γ 2 : w k = γ k + O( x 2 ), k = 0, 1, 2. If u(x) has a discontinuity in the substencil S k, then the corresponding w k is very small: w k = O( x 4 )

4. A robust choice of the nonlinear weights, given in (Jiang and Shu, JCP 1996) and used in most WENO literature, is w k = w k w 0 + w 1 + w 2, w k = γ k (ε + β k ) 2 where ε = 10 6 typically (it can be adjusted by the average size of the solution), and the smoothness indicator β k measures the smoothness of the function u(x) in the substencil S k and is given by β k = x i xi+1 2 x i 1 2 (p r(x)) 2 dx + x 3 i xi+ 1 2 x i 1 2 (p r(x)) 2 dx.

These smoothness indicators can be worked out explicitly as β 0 = 13 12 (ū i 2 2ū i 1 + ū i ) 2 + 1 4 (ū i 2 4ū i 1 + 3ū i ) 2 β 1 = 13 12 (ū i 1 2ū i + ū i+1 ) 2 + 1 4 (ū i 1 ū i+1 ) 2 β 2 = 13 12 (ū i 2ū i+1 + ū i+2 ) 2 + 1 4 (3ū i 4ū i+1 + ū i+2 ) 2

Bound-preserving limiter For many physical problems, there are natural bounds for the solution. For example, if the solution represents a percentage of a component in a mixture, then it must be between 0 and 1. If the solution is the probability density function, it must be non-negative. For Euler equations of gas dynamics, the density and pressure must be non-negative. For shallow water equations, the water height should be non-negative, etc.

We take the scalar conservation laws u t + F(u) = 0, u(x, 0) = u 0 (x) (3) as an example. An important property of the entropy solution (which may be discontinuous) is that it satisfies a strict maximum principle: If M = max x u 0 (x), m = min x u 0 (x), (4) then u(x,t) [m,m] for any x and t.

First order monotone schemes can maintain the maximum principle. For the one-dimensional conservation law the first order monotone scheme u t + f(u) x = 0, u n+1 j = H λ (u n j 1,u n j,u n j+1) = u n j λ[h(u n j,u n j+1) h(u n j 1,u n j )] where λ = t x and h(u,u + ) is a monotone flux (h(, )), satisfies under a suitable CFL condition H λ (,, ) λ λ 0.

Therefore, if m u n j 1,u n j,u n j+1 M then and u n+1 j = H λ (u n j 1,u n j,u n j+1) H λ (m,m,m) = m, u n+1 j = H λ (u n j 1,u n j,u n j+1) H λ (M,M,M) = M.

However, for higher order linear schemes, i.e. schemes which are linear for a linear PDE u t + au x = 0 for example the second order accurate Lax-Wendroff scheme u n+1 j = aλ 2 (1 + aλ)un j 1 + (1 a 2 λ 2 )u n j aλ 2 (1 aλ)un j+1 where λ = t x and a λ 1, the maximum principle is not satisfied. In fact, no linear schemes with order of accuracy higher than one can satisfy the maximum principle (Godunov Theorem).

Therefore, nonlinear schemes, namely schemes which are nonlinear even for linear PDEs, have been designed to overcome this difficulty. These include roughly two classes of schemes: TVD schemes. Most TVD (total variation diminishing) schemes also satisfy strict maximum principle, even in multi-dimensions. TVD schemes can be designed for any formal order of accuracy for solutions in smooth, monotone regions. However, all TVD schemes will degenerate to first order accuracy at smooth extrema. TVB schemes, ENO schemes, WENO schemes. These schemes do not insist on strict TVD properties, therefore they do not satisfy strict maximum principles, although they can be designed to be arbitrarily high order accurate for smooth solutions.

Remark: If we insist on the maximum principle interpreted as m u n+1 j M, j if m u n j M, j, where u n j is either the approximation to the point value u(x j,t n ) for a xj+1/2 x j 1/2 u(x,t n )dx for finite difference scheme, or to the cell average 1 x a finite volume or DG scheme, then the scheme can be at most second order accurate (proof due to Harten, see Zhang and Shu, Proceedings of the Royal Society A, 2011).

Therefore, the correct procedure to follow in designing high order schemes that satisfy a strict maximum principle is to change the definition of maximum principle. Note that a high order finite volume scheme has the following algorithm flowchart: (1) Given {ū n j } (2) reconstruct u n (x) (piecewise polynomial with cell average ū n j ) (3) evolve by, e.g. Runge-Kutta time discretization to get {ū n+1 j } (4) return to (1)

Therefore, instead of requiring m ū n+1 j M, j if we will require if m ū n j M, m u n+1 (x) M, m u n (x) M, j, x x. Similar definition and procedure can be used for discontinuous Galerkin schemes.

Maximum-principle-preserving for scalar equations The flowchart for designing a high order scheme which obeys a strict maximum principle is as follows: 1. Start with u n (x) which is high order accurate u(x,t n ) u n (x) C x p and satisfy m u n (x) M, x therefore of course we also have m ū n j M, j.

2. Evolve for one time step to get m ū n+1 j M, j. (5) 3. Given (5) above, obtain the reconstruction u n+1 (x) which satisfies the maximum principle m u n+1 (x) M, x; is high order accurate u(x,t n+1 ) u n+1 (x) C x p.

Three major difficulties 1. The first difficulty is how to evolve in time for one time step to guarantee m ū n+1 j M, j. (6) This is very difficult to achieve. Previous works use one of the following two approaches:

Use exact time evolution. This can guarantee m ū n+1 j M, j. However, it can only be implemented with reasonable cost for linear PDEs, or for nonlinear PDEs in one dimension. This approach was used in, e.g., Jiang and Tadmor, SISC 1998; Liu and Osher, SINUM 1996; Sanders, Math Comp 1988; Qiu and Shu, SINUM 2008; Zhang and Shu, SINUM 2010; to obtain TVD schemes or maximum-principle-preserving schemes for linear and nonlinear PDEs in one dimension or for linear PDEs in multi-dimensions, for second or third order accurate schemes.

Use simple time evolution such as SSP Runge-Kutta or multi-step methods. However, additional limiting will be needed on u n (x) which will destroy accuracy near smooth extrema. We have figured out a way to obtain m ū n+1 j M, j with simple Euler forward or SSP Runge-Kutta or multi-step methods without losing accuracy on the limited u n (x):

The evolution of the cell average for a higher order finite volume or DG scheme satisfies ū n+1 j = G(ū n j,u,u +,u,u + j 1 j 1 j+ 1 j+ 1 2 2 2 2) = ū n j λ[h(u,u + ) h(u j+ 1 j+ 1 j 1 2 2 2,u + j 1 2)], where G(,,,, ) therefore there is no maximum principle. The problem is with the two arguments u + j 1 and u j+ 1 which are values at points inside the cell 2 2 I j.

The polynomial p j (x) (either reconstructed in a finite volume method or evolved in a DG method) is of degree k, defined on I j such that ū n j is its cell average on I j, u + = p j 1 j (x j 1 and u 2 2) = p j+ 1 j (x j+ 1). 2 2 We take a Legendre Gauss-Lobatto quadrature rule which is exact for polynomials of degree k, then ū n j = m l=0 ω l p j (y l ) with y 0 = x j 1, y m = x 2 j+ 1. The scheme for the cell average is then 2 rewritten as

ū n+1 j = ω m [ u j+ 1 2 +ω 0 [ u + j 1 2 + m 1 l=1 λ ) (h(u ],u + ) h(u +,u ω j+ 1 j+ 1 j 1 j+ 1 m 2 2 2 2) λ ( ) ] h(u +,u ) h(u,u + ω j 1 j+ 1 j 1 j 1 0 2 2 2 2) ω l p j (y l ) = H λ/ωm (u + j 1 2 + m 1 l=1 ω l p j (y l ).,u,u + j+ 1 j+ 1 2 2 ) + H λ/ω0 (u j 1 2,u + j 1 2,u j+ 1 2)

Therefore, if m p j (y l ) M at all Legendre Gauss-Lobatto quadrature points and a reduced CFL condition λ/ω m = λ/ω 0 λ 0 is satisfied, then m ū n+1 j M.

2. The second difficulty is: given m ū n+1 j M, j how to obtain an accurate reconstruction u n+1 (x) which satisfy m u n+1 (x) M, x. Previous work was mainly for relatively lower order schemes (second or third order accurate), and would typically require an evaluation of the extrema of u n+1 (x), which, for a piecewise polynomial of higher degree, is quite costly. We have figured out a way to obtain such reconstruction with a very simple scaling limiter, which only requires the evaluation of u n+1 (x) at certain pre-determined quadrature points and does not destroy accuracy:

We replace p j (x) by the limited polynomial p j (x) defined by p j (x) = θ j (p j (x) ū n j ) + ū n j where θ j = min { M ū n j M j ū n j, m ū n j m j ū n j }, 1, with M j = max x S j p j (x), m j = min x S j p j (x) where S j is the set of Legendre Gauss-Lobatto quadrature points of cell I j. Clearly, this limiter is just a simple scaling of the original polynomial around its average.

The following lemma, guaranteeing the maintenance of accuracy of this simple limiter, is proved in Zhang and Shu, JCP 2010a: Lemma: Assume ū n j [m,m] and p j (x) is an O( x p ) approximation, then p j (x) is also an O( x p ) approximation.

3. The third difficulty is how to generalize the algorithm and result to 2D (or higher dimensions). Algorithms which would require an evaluation of the extrema of the reconstructed polynomials u n+1 (x,y) would not be easy to generalize at all. Our algorithm uses only explicit Euler forward or SSP (also called TVD) Runge-Kutta or multi-step time discretizations, and a simple scaling limiter involving just evaluation of the polynomial at certain quadrature points, hence easily generalizes to 2D or higher dimensions on structured or unstructured meshes, with strict maximum-principle-satisfying property and provable high order accuracy.

The technique has been generalized to the following situations maintaining uniformly high order accuracy: 2D scalar conservation laws on rectangular or triangular meshes with strict maximum principle (Zhang and Shu, JCP 2010a; Zhang, Xia and Shu, JSC 2012). 2D incompressible equations in the vorticity-streamfunction formulation (with strict maximum principle for the vorticity), and 2D passive convections in a divergence-free velocity field, i.e. ω t + (uω) x + (vω) x = 0, with a given divergence-free velocity field (u, v), again with strict maximum principle (Zhang and Shu, JCP 2010a; Zhang, Xia and Shu, JSC 2012).

The framework of establishing maximum-principle-satisfying schemes for scalar equations can be generalized to hyperbolic systems to preserve the positivity of certain physical quantities, such as density and pressure of compressible gas dynamics. Positivity-preserving finite volume or DG schemes have been designed for: One and multi-dimensional compressible Euler equations maintaining positivity of density and pressure (Zhang and Shu, JCP 2010b; Zhang, Xia and Shu, JSC 2012). One and two-dimensional shallow water equations maintaining non-negativity of water height and well-balancedness for problems with dry areas (Xing, Zhang and Shu, Advances in Water Resources 2010; Xing and Shu, Advances in Water Resources 2011).

One and multi-dimensional compressible Euler equations with source terms (geometric, gravity, chemical reaction, radiative cooling) maintaining positivity of density and pressure (Zhang and Shu, JCP 2011). One and multi-dimensional compressible Euler equations with gaseous detonations maintaining positivity of density, pressure and reactant mass fraction, with a new and simplified implementation of the pressure limiter. DG computations are stable without using the TVB limiter (Wang, Zhang, Shu and Ning, JCP 2012). A minimum entropy principle satisfying high order scheme for gas dynamics equations (Zhang and Shu, Num Math 2012).

A simple WENO limiter for DG methods Finite volume WENO schemes involve rather complicated reconstruction procedure for unstructured meshes. However, they have the advantage of essentially non-oscillatory performance for solutions with strong shocks. Therefore, the WENO methodology is often used as limiters for discontinuous Galerkin (DG) methods (Qiu and Shu, JCP 2003; SISC 2005; Computers & Fluids 2005; Zhu, Qiu, Shu and Dumbser, JCP 2008; Zhu and Qiu, JCP 2012.).

In particular, the very recent work in Zhong and Shu, JCP 2012; Zhu, Zhong, Shu and Qiu, JCP submitted contains a very simple and effective WENO limiter: Use a troubled-cell indicator to identify troubled cells. Qiu and Shu, SISC 2005. If the cell I j is identified as a troubled cell, then the DG solution polynomial p j (x) is replaced by a convex combination of p j (x) with p j 1 (x) and p j+1 (x), the DG solution polynomials of the two immediate neighboring cells. Suitable adjustment is made (a constant is added to p j 1 (x) to obtain p j 1 (x), likewise for p j+1 (x)) to ensure that the new polynomial maintains the original cell average (conservation).

Details: p new j = w 1 p j 1 (x) + w 2 p j (x) + w 3 p j+1 (x) where w l = with the linear weights given by w l w 1 + w 2 + w 3 ; w l = γ 1 = γ 3 = 1 1000, γ 2 = 998 1000 γ l (s l + ε) 2 and the s l are the standard smoothness indicators of WENO approximations.

Example 1: A Mach 3 wind tunnel with a step. The wind tunnel is 1 length unit wide and 3 length units long. The step is 0.2 length units high and is located 0.6 length units from the left-hand end of the tunnel. The problem is initialized by a right-going Mach 3 flow. Reflective boundary conditions are applied along the wall of the tunnel and inflow/outflow boundary conditions are applied at the entrance/exit.

1 0.9 0.8 0.7 0.6 Y 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 X Figure 3: Forward step problem. Sample mesh. The mesh points on the boundary are uniformly distributed with cell length h = 1/20.

1 Y 0.5 0 0 1 2 3 X Figure 4: Forward step problem. Third order (k = 2) RKDG with the WENO limiter. 30 equally spaced density contours from 0.32 to 6.15. The mesh points on the boundary are uniformly distributed with cell length h = 1/100.

1 Y 0.5 0 0 1 2 3 X Figure 5: Forward step problem. Third order (k = 2) RKDG with the WENO limiter. Troubled cells. Circles denote triangles which are identified as troubled cell subject to the WENO limiting. The mesh points on the boundary are uniformly distributed with cell length h = 1/100.

Example 2: We consider inviscid Euler transonic flow past a single NACA0012 airfoil configuration with Mach number M = 0.85, angle of attack α = 1. The computational domain is [ 15, 15] [ 15, 15].

1.5 1 0.5 Y/C 0-0.5-1 -1 0 1 2 X/C Figure 6: NACA0012 airfoil mesh zoom in.

1.5 1.5 1 1 0.5 0.5 Y/C 0 Y/C 0-0.5-0.5-1 -1-1.5-1.5-2 -1 0 1 2 X/C -2-1 0 1 2 X/C Figure 7: NACA0012 airfoil. Mach number. M = 0.85, angle of attack α = 1, 30 equally spaced mach number contours from 0.158 to 1.357. Left: second order (k = 1); right: third order (k = 2) RKDG with the WENO limiter.

2 2 1.5 1.5 1 1 0.5 0.5 Y/C 0 Y/C 0-0.5-0.5-1 -1-1.5-1.5-2 -1-0.5 0 0.5 1 1.5 2 X/C -2-1 -0.5 0 0.5 1 1.5 2 X/C Figure 8: NACA0012 airfoil. Troubled cells. Circles denote triangles which are identified as troubled cells subject to the WENO limiting. M = 0.85, angle of attack α = 1. Left: second order (k = 1); right: third order (k = 2) RKDG with the WENO limiter.

Concluding remarks Finite volume schemes can maintain conservation and achieve high order accuracy both for structured and unstructured meshes. WENO reconstruction in finite volume schemes can provide high order accuracy and essentially non-oscillatory shock transition. Bound-preserving limiter based on a simple scaling limiter can guarantee maximum-principle for scalar equations and passive-convection in a divergence-free velocity field, positivity for density and pressure for Euler equations, and positivity for water height for shallow water equations, among many others applications, without compromising high order accuracy.

A simple WENO limiter can be designed for discontinuous Galerkin method to handle strong shocks without affecting high order accuracy.

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