A Central Compact-Reconstruction WENO Method for Hyperbolic Conservation Laws

A Central Compact-Reconstruction WENO Method for Hyperbolic Conservation Laws Kilian Cooley 1 Prof. James Baeder 2 1 Department of Mathematics, University of Maryland - College Park 2 Department of Aerospace Engineering, University of Maryland - College Park January 8, 2018

Overview 1 WENO, CRWENO, and Central Schemes 2 Motivation 3 1D CCRWENO Scheme 4 2D Extension 5 Numerical Tests 6 Conclusion

WENO Reconstruction Weighted Essentially Non-Oscillatory Use a convex combination of linear subschemes: u j+1/2 = i ω i u (i) j+1/2

WENO Reconstruction Weighted Essentially Non-Oscillatory Use a convex combination of linear subschemes: u j+1/2 = i ω i u (i) j+1/2 (1) u (1) j+1/2 = 1 7 11 3ūj 2 6ūj 1+ 6 ūj ω 1 = 1 10 u (2) j+1/2 = 1 5 6ūj 1+ + 1 ω 2 = 3 6ūj 3ūj+1 5 u (3) j+1/2 = 1 + 5 1 ω 3 = 3 3ūj 6ūj+1 6ūj+2 10

WENO Reconstruction Weighted Essentially Non-Oscillatory Use a convex combination of linear subschemes: u j+1/2 = i ω i u (i) j+1/2 (1) u (1) j+1/2 = 1 7 11 3ūj 2 6ūj 1+ 6 ūj ω 1 = 1 10 u (2) j+1/2 = 1 5 6ūj 1+ + 1 ω 2 = 3 6ūj 3ūj+1 5 u (3) j+1/2 = 1 + 5 1 ω 3 = 3 3ūj 6ūj+1 6ūj+2 10 u is smooth in all stencils ω i ω i and Eq. (1) is O ( x 5).

WENO Reconstruction Weighted Essentially Non-Oscillatory Use a convex combination of linear subschemes: u j+1/2 = i ω i u (i) j+1/2 (1) u (1) j+1/2 = 1 7 11 3ūj 2 6ūj 1+ 6 ūj ω 1 = 1 10 u (2) j+1/2 = 1 5 6ūj 1+ + 1 ω 2 = 3 6ūj 3ūj+1 5 u (3) j+1/2 = 1 + 5 1 ω 3 = 3 3ūj 6ūj+1 6ūj+2 10 u is smooth in all stencils ω i ω i and Eq. (1) is O ( x 5). u is not smooth in stencil i ω i 0.

Non-Oscillatory Weights Produce this behavior by scaling the ideal weights ω i according to smoothness indicators β i [Jiang & Shu, 1996]: β 1 = 13 12 (u j 2 2u j 1 + u j ) 2 + 1 4 (u j 2 4u j 1 + 3u j ) 2 β 2 = 13 12 (u j 1 2u j + u j+1 ) 2 + 1 4 (u j 1 u j+1 ) 2 β 3 = 13 12 (u j 2u j+1 + u j+2 ) 2 + 1 4 (3u j 4u j+1 + u j+2 ) 2 ω i α i = (ɛ + β i ) 2 ω i = α i m α m

CRWENO Schemes Ghosh and Baeder [Ghosh & Baeder, 2012] used compact subschemes to produce the Compact-Reconstruction WENO scheme: 2 3 u j 1/2+ 1 3 u j+1/2 = 1 5 6ūj 1+ 6ūj ω 1 = 1 5 1 3 u j 1/2+ 2 3 u j+1/2 = 5 + 1 6ūj 6ūj+1 ω 2 = 1 2 2 3 u j+1/2 + 1 3 u j+3/2 = 1 + 5 6ūj 6ūj+1 ω 3 = 3 10 Solve a tridiagonal system for the interface values.

CRWENO Schemes Ghosh and Baeder [Ghosh & Baeder, 2012] used compact subschemes to produce the Compact-Reconstruction WENO scheme: 2 3 u j 1/2+ 1 3 u j+1/2 = 1 5 6ūj 1+ 6ūj ω 1 = 1 5 1 3 u j 1/2+ 2 3 u j+1/2 = 5 + 1 6ūj 6ūj+1 ω 2 = 1 2 2 3 u j+1/2 + 1 3 u j+3/2 = 1 + 5 6ūj 6ūj+1 ω 3 = 3 10 Solve a tridiagonal system for the interface values. CRWENO has smaller truncation error than WENO.

CRWENO Schemes Ghosh and Baeder [Ghosh & Baeder, 2012] used compact subschemes to produce the Compact-Reconstruction WENO scheme: 2 3 u j 1/2+ 1 3 u j+1/2 = 1 5 6ūj 1+ 6ūj ω 1 = 1 5 1 3 u j 1/2+ 2 3 u j+1/2 = 5 + 1 6ūj 6ūj+1 ω 2 = 1 2 2 3 u j+1/2 + 1 3 u j+3/2 = 1 + 5 6ūj 6ūj+1 ω 3 = 3 10 Solve a tridiagonal system for the interface values. CRWENO has smaller truncation error than WENO. Implicitness in space improves resolution of small-scale features.

Extension to Systems WENO and CRWENO reconstructions fit into the Godunov framework: reconstruct left- and right-biased interface values and use a Riemann solver. Oscillations can arise if reconstructions are not performed in characteristic variables ξ = Cu, where C is the matrix of left eigenvectors of the flux Jacobian.

Extension to Systems WENO and CRWENO reconstructions fit into the Godunov framework: reconstruct left- and right-biased interface values and use a Riemann solver. Oscillations can arise if reconstructions are not performed in characteristic variables ξ = Cu, where C is the matrix of left eigenvectors of the flux Jacobian. The tridiagonal system of CRWENO becomes a block-tridiagonal system.

Extension to Systems WENO and CRWENO reconstructions fit into the Godunov framework: reconstruct left- and right-biased interface values and use a Riemann solver. Oscillations can arise if reconstructions are not performed in characteristic variables ξ = Cu, where C is the matrix of left eigenvectors of the flux Jacobian. The tridiagonal system of CRWENO becomes a block-tridiagonal system. CRWENO with characteristic variables becomes less efficient than WENO with characteristic variables.

Extension to Systems WENO and CRWENO reconstructions fit into the Godunov framework: reconstruct left- and right-biased interface values and use a Riemann solver. Oscillations can arise if reconstructions are not performed in characteristic variables ξ = Cu, where C is the matrix of left eigenvectors of the flux Jacobian. The tridiagonal system of CRWENO becomes a block-tridiagonal system. CRWENO with characteristic variables becomes less efficient than WENO with characteristic variables. M1: Can we avoid using characteristic variables?

Extension to Systems WENO and CRWENO reconstructions fit into the Godunov framework: reconstruct left- and right-biased interface values and use a Riemann solver. Oscillations can arise if reconstructions are not performed in characteristic variables ξ = Cu, where C is the matrix of left eigenvectors of the flux Jacobian. The tridiagonal system of CRWENO becomes a block-tridiagonal system. CRWENO with characteristic variables becomes less efficient than WENO with characteristic variables. M1: Can we avoid using characteristic variables? M2: Can we eliminate the Riemann solver, and does doing so achieve M1?

Central Schemes A Riemann solver is needed to define a flux where the reconstruction is discontinuous. Central schemes avoid this by introducing a staggered grid: Staggered grid Main grid x j x j 1/2 x j+1/2

Central Schemes A Riemann solver is needed to define a flux where the reconstruction is discontinuous. Central schemes avoid this by introducing a staggered grid: Staggered grid Main grid x j x j 1/2 x j+1/2 Applying the integral form of a conservation law to a staggered cell gives: u (u) tn+1 + f = 0 ū n+1 t x j+1/2 = ūn j+1/2 f (u(x j+1 )) f (u(x j )) dx t n x

Time Advancement In Central Schemes Evaluate the time integrals by quadrature (Simpson s rule). Evolve the point values using Runge-Kutta with the differential form. Sample a natural continuous extension at quadrature nodes.

Time Advancement In Central Schemes RK4: Evaluate the time integrals by quadrature (Simpson s rule). Evolve the point values using Runge-Kutta with the differential form. Sample a natural continuous extension at quadrature nodes. u n+1 = u n + t 4 b k G k k=1

Time Advancement In Central Schemes RK4: Evaluate the time integrals by quadrature (Simpson s rule). Evolve the point values using Runge-Kutta with the differential form. Sample a natural continuous extension at quadrature nodes. u n+1 = u n + t 4 b k G k k=1 Natural Continuous Extension [Zennaro, 1986]: z(t n + θ t) = u n + t 4 b k (θ)g k = u(t) + O ( t 4) k=1 z(t n ) = u n and z(t n+1 ) = u n+1.

Central Scheme Steps 1 From cell averages on the main grid at time index n, reconstruct left-subcell averages. 2 Compute right-subcell averages by conservation. 3 Compute averages over staggered cells. 4 Reconstruct point values at staggered-cell interfaces (i.e., main-cell midpoints). 5 Compute fluxes and flux derivatives at staggered-cell interfaces. 6 Repeat step 5 as needed to calculate the time integrals in the update equation. 7 Compute staggered-cell averages at time index n + 1.

Central WENO Schemes Scheme Order Reconstruction Characteristic approach variables? [Levy, Puppo, & Russo 2002] 4th Polynomials No CWENO5 [Qiu & Shu, 2002] 5th FV formulas Yes, for SA

Central WENO Schemes Scheme Order Reconstruction Characteristic approach variables? [Levy, Puppo, & Russo 2002] 4th Polynomials No CWENO5 [Qiu & Shu, 2002] 5th FV formulas Yes, for SA Observation: The polynomial approach guarantees identical weights for the subcell-average and point-value reconstructions. Those weights do not match in CWENO5.

Central WENO Schemes Scheme Order Reconstruction Characteristic approach variables? [Levy, Puppo, & Russo 2002] 4th Polynomials No CWENO5 [Qiu & Shu, 2002] 5th FV formulas Yes, for SA Observation: The polynomial approach guarantees identical weights for the subcell-average and point-value reconstructions. Those weights do not match in CWENO5. M3: Would a central WENO scheme using the same weights for these reconstructions still require characteristic variables?

Motivation 1 Restore the advantage of compact reconstructions for systems by avoiding the need for characteristic variables.

Motivation 1 Restore the advantage of compact reconstructions for systems by avoiding the need for characteristic variables. 2 Eliminate the Riemann solver.

Motivation 1 Restore the advantage of compact reconstructions for systems by avoiding the need for characteristic variables. 2 Eliminate the Riemann solver. 3 Test the significance of identical weights for subcell-average and point-value reconstructions.

Constraints On The CCRWENO Subschemes

Constraints On The CCRWENO Subschemes 1 SA and PV reconstructions must use the same ideal weights.

Constraints On The CCRWENO Subschemes 1 SA and PV reconstructions must use the same ideal weights. Normalization: 2 Ideal weights must be positive and sum to 1. 3 LHS coefficients must sum to 1 for all weight combinations.

Constraints On The CCRWENO Subschemes 1 SA and PV reconstructions must use the same ideal weights. Normalization: 2 Ideal weights must be positive and sum to 1. 3 LHS coefficients must sum to 1 for all weight combinations. Accuracy: 4 SA and PV reconstructions must be at least fifth-order accurate.

Constraints On The CCRWENO Subschemes 1 SA and PV reconstructions must use the same ideal weights. Normalization: 2 Ideal weights must be positive and sum to 1. 3 LHS coefficients must sum to 1 for all weight combinations. Accuracy: 4 SA and PV reconstructions must be at least fifth-order accurate. Stability: 5 LHS coefficient matrix must be diagonally dominant for all weight combinations.

Constraints On The CCRWENO Subschemes 1 SA and PV reconstructions must use the same ideal weights. Normalization: 2 Ideal weights must be positive and sum to 1. 3 LHS coefficients must sum to 1 for all weight combinations. Accuracy: 4 SA and PV reconstructions must be at least fifth-order accurate. Stability: 5 LHS coefficient matrix must be diagonally dominant for all weight combinations. Symmetry: 6 Ideal weights must be symmetric: ω 1 = ω 3. 7 LHS of the combined scheme with ideal weights must be symmetric.

Identical Coefficient Matrices If we use the same smoothness indicators for SA and PV reconstructions, we can merge them into one linear system by designing their subschemes so that the two coefficient matrices are identical. 8 LHS coefficients of each SA subscheme must match those of the corresponding PV subscheme.

Identical Coefficient Matrices If we use the same smoothness indicators for SA and PV reconstructions, we can merge them into one linear system by designing their subschemes so that the two coefficient matrices are identical. 8 LHS coefficients of each SA subscheme must match those of the corresponding PV subscheme. A one-parameter family of such schemes exists that satisfy all eight conditions.

The CCRWENO Subschemes: SA and PV LHS LHS coefficients of subschemes: L = 1 d 1 2 1 d 2 4 0 1+d 1 2 0 1+d 2 1 d 2 2 4 1+d 1 1 d 1 2 2

The CCRWENO Subschemes: SA and PV LHS LHS coefficients of subschemes: L = 1 d 1 2 1 d 2 4 0 1+d 1 2 0 1+d 2 1 d 2 2 4 1+d 1 1 d 1 2 2 d 1 > 0, d 2 = 5 + 8d 1 17 + 20d 1 ω 1 = 1 ω 2, ω 2 = d 1 + 1 4, ω 3 = 1 ω 2 2 d 1 + d 2 2

The CCRWENO Subschemes: SA and PV RHS ū L j 1 d 1 16 3 d 1 8 3 2d 2 16 L ūj L = 0 ūj+1 L 0 0 u j 1 1 24 7 6d 1 12 5 6d 2 L u j = 0 24 u j+1 0 0 16 0 0 1 2d 2 4 16 0 3d 1 +2 1+d 2 5d 1 +6 16 11+12d 1 1 3d 1 8 d 1 16 24 0 0 5 6d 2 12 24 0 7+6d 2 11+12d 1 24 7 6d 1 12 1 24 ū j 2 ū j 1 ū j ū j+1 ū j+2 ū j 2 ū j 1 ū j ū j+1 ū j+2

The CCRWENO Subschemes: SA and PV RHS ū L j 1 d 1 16 3 d 1 8 3 2d 2 16 L ūj L = 0 ūj+1 L 0 0 u j 1 1 24 7 6d 1 12 5 6d 2 L u j = 0 24 u j+1 0 0 16 0 0 1 2d 2 4 16 0 3d 1 +2 1+d 2 5d 1 +6 16 11+12d 1 1 3d 1 8 d 1 16 24 0 0 5 6d 2 12 24 0 7+6d 2 11+12d 1 24 7 6d 1 12 1 24 ū j 2 ū j 1 ū j ū j+1 ū j+2 ū j 2 ū j 1 ū j ū j+1 ū j+2 The PV reconstruction is unstable for d 1 1.2. All results will use d 1 = 1.3.

The CCRWENO Subschemes: FD Reconstruction Each time step involves four FD reconstructions.

The CCRWENO Subschemes: FD Reconstruction Each time step involves four FD reconstructions. Cannot reuse SA/PV coefficient matrix because either the weights or the LHS coefficients must be different in order to satisfy the accuracy conditions.

The CCRWENO Subschemes: FD Reconstruction Each time step involves four FD reconstructions. Cannot reuse SA/PV coefficient matrix because either the weights or the LHS coefficients must be different in order to satisfy the accuracy conditions. No symmetric and diagonally dominant scheme with positive weights exceeds the fourth-order accuracy provided by: x(u x ) j = 1 2 u j 2 2u j 1 + 3 2 u j ω 1 = 1 6 x(u x ) j = 1 2 u j 1 + 2u j + 1 2 u j+1 ω 2 = 2 3 x(u x ) j = 3 2 u j + 2u j+1 1 2 u j+2 ω 3 = 1 6 We use these subschemes for the flux derivative reconstruction.

2D Staggered Grid Main grid Staggered grid Integrate fluxes over faces by quadrature.

Outer Product Extension Compose a 1D subscheme r in the x direction with a 1D subscheme s in the y direction. Order of accuracy is preserved. x: L r mūi+m L = m Rr mū i+m m y: L s nūj+n L = n Rs nū j+n n m,n L r ml s nū LL i+m,j+n = m,n R r mr s nū i+m,j+n

Outer Product Extension Compose a 1D subscheme r in the x direction with a 1D subscheme s in the y direction. Order of accuracy is preserved. x: L r mūi+m L = m Rr mū i+m m y: L s nūj+n L = n Rs nū j+n n m,n L r ml s nū LL i+m,j+n = m,n R r mr s nū i+m,j+n j + 1 L r 1 Ls 1 L r 0 Ls 1 L r 1 Ls 1 j L r 1 Ls 0 L r 0 Ls 0 L r 1 Ls 0 ω r,s = ω r ω s j 1 L r 1 Ls 1 i 1 L r 0 Ls 1 i L r 1 Ls 1 i + 1

Linear Advection: Convergence Linear advection of sin 2 (πx) sin 2 (πy) with velocity (u, v) = (1, 1).

Linear Advection: Efficiency Compared to [Levy, Puppo, & Russo 2002] and the outer-product extension of CWENO5 in [Qiu & Shu, 2002].

Isentropic Vortex Advection: Convergence Smooth solution of the Euler equations.

Isentropic Vortex Advection: Efficiency

Lax Problem

Shu-Osher Problem

2D Riemann Problem: Configuration 5 Density contours for Configuration 5 in [Lax & Liu, 1998] computed with CCRWENO. 200 200 cells

2D Riemann Problem: Configuration 16 Density contours for Configuration 16 in [Lax & Liu, 1998] computed by CCRWENO. 400 400 cells

2D Riemann Problem: Configuration 16 A cross-section of the CCRWENO solution at x = 0.9 shows the small amplitude of the oscillations.

2D Riemann Problem: Configuration 16 Density contours computed by CWENO5. 400 400 cells

Advection of High-Frequency Waves Euler advection of ρ(x, y) = 2 + sin(2πkx) sin(2πky) with velocity (u, v) = (1, 0) for a range of wavenumbers k.

Conclusion Original motivations: Avoid characteristic variables and preserve advantages of compact reconstructions.

Conclusion Original motivations: Avoid characteristic variables and preserve advantages of compact reconstructions. Small oscillations appear in the Lax problem and Configuration 16. CCRWENO does resolve fine features more efficiently.

Conclusion Original motivations: Avoid characteristic variables and preserve advantages of compact reconstructions. Small oscillations appear in the Lax problem and Configuration 16. CCRWENO does resolve fine features more efficiently. Avoid Riemann solvers. 2D physics captured without special treatment.

Conclusion Original motivations: Avoid characteristic variables and preserve advantages of compact reconstructions. Small oscillations appear in the Lax problem and Configuration 16. CCRWENO does resolve fine features more efficiently. Avoid Riemann solvers. 2D physics captured without special treatment. Avoid oscillations by using identical SA and PV weights. Fewer oscillations in Configuration 16.

Conclusion Original motivations: Avoid characteristic variables and preserve advantages of compact reconstructions. Small oscillations appear in the Lax problem and Configuration 16. CCRWENO does resolve fine features more efficiently. Avoid Riemann solvers. 2D physics captured without special treatment. Avoid oscillations by using identical SA and PV weights. Fewer oscillations in Configuration 16. CCRWENO is more computationally efficient than the Levy scheme or CWENO5.

Future Work Remove oscillations in the Lax problem and Configuration 16. Implement more general boundary conditions. Extend to Navier-Stokes equations.

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