Parametrized maximum principle preserving flux limiters for high - PowerPoint PPT Presentation

Parametrized maximum principle preserving flux limiters for high order schemes: Algorithm, Development, Application Zhengfu Xu Michigan Technological University Collaborators: J.-M. Qiu, T. Xiong (U. Houston), Y. Jiang (ISU) , C. Liang (MTU), Y. Liu, Q. Tang, A. Christlieb (MSU) May 23, 2014

Outline ◮ Discrete maximum principle preserving (MPP). ◮ A sufficient condition for conservative schemes to be MPP. ◮ General MPP flux limiters ◮ Partial analysis results regarding CFL and accuracy. ◮ Applications and results. ◮ Conclusion with ongoing and future work.

Four major concerns for solving HCL u t + f ( u ) x = 0 numerically: 1. Accuracy: High order reconstruction. 2. Stability: MPP provides a weak stability. 3. Conservation: This is the basic framework for discussion. 4. Efficiency: CFL number shall not be too small.

High order finite volume and finite difference method High order finite volume formulation: integrate the PDE over I j = [ x j − 1 / 2 , x j +1 / 2 ] dt = − f j +1 / 2 − f j − 1 / 2 d ¯ u j (1) ∆ x 1 � evolving ¯ u j = I j udx . ∆ x u j → u ± 1. ¯ j +1 / 2 by polynomial reconstruction. For example, find p ( x ) of order 2 such that 1 � ¯ u k = p ( x ) dx , k = j − 1 , j , j + 1 . ∆ x I k Then let u j +1 / 2 ≈ p ( x j +1 / 2 ). 2. u ± j +1 / 2 → f j +1 / 2 by applying suitable approximate flux function.

High order finite difference method 1. Finite difference formulation with a sliding function � x + ∆ x 1 2 h ( ξ ) d ξ = f ( u ( x , t )) , ∆ x x − ∆ x 2 1 � h ( x + ∆ x 2 ) − h ( x − ∆ x � f ( u ) x = 2 ) , ∆ x du j dt = − 1 � � h j + / 2 − h j − 1 / 2 . ∆ x 2. Treat f ( u ) as the average volume of some function h , then back to the finite volume reconstruction: f ( u j ) → h ± j +1 / 2 . 3. Reconstruction: ENO (Essentially-Non-Oscillatory); WENO (Weighted ENO); or others.

MPP high order scheme for hyperbolic equation ◮ One dimensional scalar hyperbolic problem u t + f ( u ) x = 0 , u ( x , 0) = u 0 ( x ) , (2) with boundary condition. MPP means: u m ≤ u ( x , t ) ≤ u M if u m = min x u 0 ( x ) and u M = max x u 0 ( x )

MPP high order scheme for hyperbolic equation ◮ One dimensional scalar hyperbolic problem u t + f ( u ) x = 0 , u ( x , 0) = u 0 ( x ) , (2) with boundary condition. MPP means: u m ≤ u ( x , t ) ≤ u M if u m = min x u 0 ( x ) and u M = max x u 0 ( x ) ◮ A typical conservative scheme with Euler forward j − λ (ˆ H j +1 / 2 − ˆ u n +1 = u n H j − 1 / 2 ) . (3) j

MPP high order scheme for hyperbolic equation ◮ One dimensional scalar hyperbolic problem u t + f ( u ) x = 0 , u ( x , 0) = u 0 ( x ) , (2) with boundary condition. MPP means: u m ≤ u ( x , t ) ≤ u M if u m = min x u 0 ( x ) and u M = max x u 0 ( x ) ◮ A typical conservative scheme with Euler forward u n +1 j − λ (ˆ H j +1 / 2 − ˆ = u n H j − 1 / 2 ) . (3) j ◮ The goal: To achieve MPP numerically in high order, conservative manner: u m ≤ u n j ≤ u M

MPP finite volume scheme by Zhang & Shu ◮ Numerical formulation: u n +1 j +1 / 2 , u + j − 1 / 2 , u + u n j − λ { F ( u − j +1 / 2 ) − F ( u − ¯ = ¯ j − 1 / 2 ) } . j

MPP finite volume scheme by Zhang & Shu ◮ Numerical formulation: u n +1 j +1 / 2 , u + j − 1 / 2 , u + u n j − λ { F ( u − j +1 / 2 ) − F ( u − ¯ = ¯ j − 1 / 2 ) } . j ◮ u − j +1 / 2 , u + j +1 / 2 are obtained by ENO/WENO reconstruction. u ± j +1 / 2 = p ± E / W ( x j +1 / 2 )

MPP finite volume scheme by Zhang & Shu ◮ Numerical formulation: u n +1 j +1 / 2 , u + j − 1 / 2 , u + u n j − λ { F ( u − j +1 / 2 ) − F ( u − ¯ = ¯ j − 1 / 2 ) } . j ◮ u − j +1 / 2 , u + j +1 / 2 are obtained by ENO/WENO reconstruction. u ± j +1 / 2 = p ± E / W ( x j +1 / 2 ) ◮ A sufficient condition: rescaling p ± E / W ( x ) p ± E / W ( x ) = θ ( p ± E / W ( x ) − ¯ u ) + ¯ u such that u m ≤ p ± E / W ( x α ) ≤ u M at quadratue points x α (include the end points).

Development ◮ The rescaling technique has been fitted into high order finite volume and DG schemes on various mesh structures.

Development ◮ The rescaling technique has been fitted into high order finite volume and DG schemes on various mesh structures. ◮ It has been applied to positivity preserving high order finite volume and finite difference schemes for compressible Euler equations. Zhang & Shu 2010, 2011, 2012

Development ◮ The rescaling technique has been fitted into high order finite volume and DG schemes on various mesh structures. ◮ It has been applied to positivity preserving high order finite volume and finite difference schemes for compressible Euler equations. Zhang & Shu 2010, 2011, 2012 ◮ The nonexistence of high order MPP finite difference scheme is the motivation of this work.

The flow chart of high order method ◮ Reconstruction → Numerical flux → Evolution: Runge-Kutta

The flow chart of high order method ◮ Reconstruction → Numerical flux → Evolution: Runge-Kutta ↑ ↑ Zhang & Shu rescaling MPP

The flow chart of high order method ◮ Reconstruction → Numerical flux → Evolution: Runge-Kutta ↑ ↑ Zhang & Shu rescaling MPP ◮ Difficult to generalize to high order finite difference scheme: no reference value of ˆ H j +1 / 2 for rescaling p ( x ).

The flow chart of high order method ◮ Reconstruction → Numerical flux → Evolution: Runge-Kutta ↑ ↑ Zhang & Shu rescaling MPP ◮ Difficult to generalize to high order finite difference scheme: no reference value of ˆ H j +1 / 2 for rescaling p ( x ). An alternative: ◮ Reconstruction → Numerical flux → Evolution ↑ Parametrized flux limiters

Parametrized flux limiters Looking for limiters of the type H j +1 / 2 = θ j +1 / 2 (ˆ ˜ H j +1 / 2 − ˆ h j +1 / 2 ) + ˆ h j +1 / 2 (4) such that j − λ (˜ H j +1 / 2 − ˜ u m ≤ u n H j − 1 / 2 ) ≤ u M . (5) ˆ h j +1 / 2 is the first order monotone flux.

Parametrized flux limiters Looking for limiters of the type H j +1 / 2 = θ j +1 / 2 (ˆ ˜ H j +1 / 2 − ˆ h j +1 / 2 ) + ˆ h j +1 / 2 (4) such that j − λ (˜ H j +1 / 2 − ˜ u m ≤ u n H j − 1 / 2 ) ≤ u M . (5) ˆ h j +1 / 2 is the first order monotone flux. ◮ θ j +1 / 2 = 0 , j = 1 , 2 , 3 , ... : first order scheme with MPP property.

Parametrized flux limiters Looking for limiters of the type H j +1 / 2 = θ j +1 / 2 (ˆ ˜ H j +1 / 2 − ˆ h j +1 / 2 ) + ˆ h j +1 / 2 (4) such that j − λ (˜ H j +1 / 2 − ˜ u m ≤ u n H j − 1 / 2 ) ≤ u M . (5) ˆ h j +1 / 2 is the first order monotone flux. ◮ θ j +1 / 2 = 0 , j = 1 , 2 , 3 , ... : first order scheme with MPP property. ◮ θ j +1 / 2 = 1 , j = 1 , 2 , 3 , ... : high order scheme most likely without MPP property.

Parametrized flux limiters Looking for limiters of the type H j +1 / 2 = θ j +1 / 2 (ˆ ˜ H j +1 / 2 − ˆ h j +1 / 2 ) + ˆ h j +1 / 2 (4) such that j − λ (˜ H j +1 / 2 − ˜ u m ≤ u n H j − 1 / 2 ) ≤ u M . (5) ˆ h j +1 / 2 is the first order monotone flux. ◮ θ j +1 / 2 = 0 , j = 1 , 2 , 3 , ... : first order scheme with MPP property. ◮ θ j +1 / 2 = 1 , j = 1 , 2 , 3 , ... : high order scheme most likely without MPP property. ◮ θ j +1 / 2 exists, locally defined. [Xu, Math Comp 2013], online. [Liang & Xu, JSC 2014].

Comments ◮ Early work of flux limiters (TVD stability): [P L Roe, 1982; Van Leer, 1974; R. F. Warming AND R. M. Beam, 1976; P K Sweby, 1984] ◮ The parametrized limiters provide a sufficient condition for high order conservative scheme to be maximum principle preserving. ◮ The accuracy of the high order scheme with MPP limiters will be affected by CFL number. ◮ Question remains: Does it produce MPP solution with high order accuracy when applied to FD ENO/WENO?

Numerical experiments Third order finite volume ENO scheme with MPP parametrized limiters Consider f ( u ) = u , u ( x , 0) = sin(2 π x ) on the interval [0 , 1] with periodic BC. Let T be 1, one period of evolution. TVD third order Runge-Kutta time discretization is used in all the tests. Table : Third order finite volume ENO scheme with parametrized limiters: advection equation CFL=0.15 CFL=0.2 CFL=0.25 N L ∞ error order L ∞ error order L ∞ error order 40 2.18E-3 2.04E-3 2.13E-3 80 2.71E-4 3.00 2.56E-4 3.00 2.75E-4 2.95 160 3.37E-5 3.00 3.19E-5 3.00 4.15E-5 2.72 320 4.18E-6 3.01 3.99E-6 3.00 8.28E-6 2.32 640 5.18E-7 3.01 4.98E-7 3.00 2.07E-6 1.99

Numerical experiments Third order finite volume scheme with MPP parametrized limiters for Burgers’ equation In the case of inviscid Burgers’ equation f ( u ) = u 2 2 with initial condition u ( x , 0) = 0 . 5(1 + sin (2 π x )) on the interval [0 , 1], let the simulation run to T = 0 . 5 π . Table : Third order finite volume scheme with parametrized limiters: Burgers’ equation CFL=0.15 CFL=0.2 CFL=0.4 N L ∞ error order L ∞ error order L ∞ error order 40 1.90E-3 1.89E-3 1.87E-3 80 2.80E-4 2.75 2.80E-4 2.75 2.79E-4 2.75 160 3.71E-5 2.91 3.71E-5 2.91 3.69E-5 2.91 320 4.67E-6 2.98 4.67E-6 2.98 9.03E-6 2.03 640 5.84E-7 3.00 5.84E-7 3.00 2.30E-6 1.97