t f a r D Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu School of Mathematical Science Xiamen University jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu Collaborators: J. Zhu (NUAA), C.-W. Shu (Brown), H. Zhu (NUPT), X. Zhong (MSU), G. Li (Qingdao), Y. Cheng (Baidu Company, Beijing), M. Dumbser (Trento) • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Outline � Introduction � Numerical Method � Numerical results � Conclusions • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Introduction 1 Introduction • We consider hyperbolic conservation laws: � u t + ∇ · f ( u ) = 0 , u ( x, 0) = u 0 ( x ) . • Hyperbolic conservation laws and convection dominated PDEs play an impor- tant role arise in applications, such as gas dynamics, modeling of shallow wa- ters,... • There are special difficulties associated with solving these systems both on mathematical and numerical methods, for discontinuous may appear in the so- lutions for nonlinear equations, even though the initial conditions are smooth enough. 1 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Introduction • This is why devising robust, accurate and efficient methods for numerically solv- ing these problems is of considerable importance and as expected, has attracted the interest of many researchers and practitioners. • Within recent decades, many high-order numerical methods have been devel- oped to solve these problem. Among them, we would like to mention Discontin- uous Galerkin (DG) method and Weighted essentially non-oscillatory (WENO) scheme. • DG method is a high order finite element method. • WENO scheme is finite difference or finite volume scheme. • Both DG and WENO are very important numerical methods for the Convection Dominated PDEs. 2 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Introduction • The first DG was presented by Reed and Hill in 1973, in the framework of neutron transport (steady state linear hyperbolic equations). • From 1987, a major development of the DG method was carried out by Cock- burn, Shu et al. in a series of papers. • They established a framework to easily solve nonlinear time dependent hyper- bolic conservation laws using explicit, nonlinearly stable high order Runge- Kutta time discretization and DG discretization in space. These methods are termed RKDG methods. • DG employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, and limiters. 3 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Introduction • Limiter is an important component of RKDG methods for solving convection dominated problems with strong shocks in the solutions, which is applied to detect discontinuities and control spurious oscillations near such discontinuities. • Many such limiters have been used in the literature on RKDG methods such as the minmod type TVB limiter by Coukburn and Shu et al. , the moment based limiter developed by Flaherty et al. . • Limiters have been an extensively studied subject for the DG methods, however it is still a challenge to find limiters which are robust, maintaining high order accuracy in smooth regions including at smooth extrema, and yielding sharp, non-oscillatory discontinuity transitions. 4 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Introduction WENO schemes have following advantages: • Uniform high order accuracy in smooth regions including 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. • The limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of WENO finite volume and finite dif- ference methods. • In this presentation , we would like to show the design of a robust limiter for the RKDG methods based on WENO methods. 5 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method 2 Numerical Methods We consider one dimensional conservation laws: u t + f ( u ) x = 0 . Let x i are the centers of the cells I i = [ x i − 1 2 , x i + 1 2 ] , ∆ x i = x i + 1 2 − x i − 1 2 , h = sup i ∆ x i . The solution and the test function space: V k h = { p : p | I i ∈ P k ( I i ) } . • A local orthogonal basis over I i , � 2 � x − x i 1 ( x ) = x − x i − 1 v ( i ) v ( i ) v ( i ) 0 ( x ) = 1 , , 2 ( x ) = 12 , · · · ∆ x i ∆ x i 6 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method • The numerical solution u h ( x, t ) : k u ( l ) i ( t ) v ( i ) � u h ( x, t ) = l ( x ) , for x ∈ I i l =0 • The degrees of freedom u ( l ) i ( t ) are the moments: i ( t ) = 1 � u ( l ) u h ( x, t ) v ( i ) l ( x ) dx, l = 0 , 1 , · · · , k a l I i I i ( v ( i ) � l ( x )) 2 dx where a l = 7 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method • In order to evolve the degrees of freedom u ( l ) i ( t ) , we time equation u t + f ( u ) x = 0 with basis v ( i ) l ( x ) , and integrate it on cell I i , using integration by part, we obtain: � d i ( t ) + 1 � f ( u h ( x, t )) d dtu ( l ) dxv ( i ) l ( x ) dx + f ( u h ( x i +1 / 2 , t )) v ( i ) − l ( x i +1 / 2 ) a l I i � − f ( u h ( x i − 1 / 2 , t )) v ( i ) l ( x i − 1 / 2 ) = 0 , l = 0 , 1 , · · · , k • However, the boundary terms f ( u i +1 / 2 ) and v i +1 / 2 etc. are not well defined when u and v are in this space, as they are discontinuous at the cell interfaces. 8 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method • From the conservation and stability (upwinding) considerations, we take – A single valued monotone numerical flux to replace f ( u i +1 / 2 ) : f i +1 / 2 = ˆ ˆ f ( u − i +1 / 2 , u + i +1 / 2 ) where ˆ f ( u ; u ) = f ( u ) (consistency); ˆ f ( ↑ , ↓ ) (monotonicity) and ˆ f is Lips- chitz continuous with respect to both arguments. – Values from inside I i for the test function v : v ( i ) i +1 / 2 ) , v ( i ) l ( x − l ( x + i − 1 / 2 ) • We get semi-discretization scheme: � d i ( t ) + 1 � f ( u h ( x, t )) d dtu ( l ) dxv ( i ) l ( x ) dx + ˆ i +1 / 2 ) v ( i ) f ( u − i +1 / 2 , u + l ( x − − i +1 / 2 ) a l I i � i +1 / 2 ) v ( i ) − ˆ f ( u − i − 1 / 2 , u − l ( x + i − 1 / 2 ) = 0 , l = 0 , 1 , · · · , k. ( ∗ ) 9 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method Using explicit, nonlinearly stable high order Runge-Kutta time discretizations. [Shu and Osher, JCP, 1988] The semidiscrete scheme ( ∗ ) is written as: u t = L ( u ) is discretized in time by a nonlinearly stable Runge-Kutta time discretization, e.g. the third order version. u (1) = u n + ∆ tL ( u n ) u (2) = 3 4 u n + 1 4 u (1) + 1 4∆ tL ( u (1) ) u n +1 = 1 3 u n + 2 3 u (2) + 2 3∆ tL ( u (2) ) . 10 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method Lax problem. t = 1 . 3 . 200 cells. Density. Left: k = 1 . Right: k = 2 . k = 3 code blows up. For Blast Wave problem, code blows up for any k . 11 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method Limiters Many limiters have been used in the literature, such as: • The minmod based TVB limiter.( Cockburn and Shu, Math. Comp. 1989) • Moment limiter. (Biswas, Devine and Flaherty, Appl. Numer. Math, 1994) • A modification of moment limiter.(Burbean, Sagaut and Brunean, JCP, 2001) • The monotonicity preserving (MP) limiter. (Suresh and Huynh, JCP, 1997) • A modification of the MP limiter. (Rider and Margolin, JCP, 2001) 12 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method These limiters tend to degrade accuracy when mistakenly used in smooth regions of the solution. Burgers equation, initial condition u ( x, 0) = 1 4 + 1 2 sin( π (2 x − 10)) , with periodic boundary condition, RKDG with TVB limiter, t=0.05. Cockburn and Shu, JSC (2001) 13 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
t f a r D Numerical Method Burbean, Sagaut and Brunean, JCP, (2001) 14 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
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