Xiaolei Chen Advisor: Prof. Xiaolin Li Department of Applied Mathematics and Statistics Stony Brook University, The State University of New York February 7, 014
1 3 Mapped WENO-Z Scheme
1D Scalar Hyperbolic Equation: u t + (f(u)) x = 0 Assume the grid is uniform and solve the hyperbolic equation directly using a conservative approximation to the spatial derivative. du i (t) dt = 1 x ( ˆf i+ 1 ˆf i 1 ), where u i (t) is the numerical approximation to the point value u(x i, t), and ˆf i+ 1 is called numerical flux. Question: How to approximate the numerical flux ˆf i+ 1?
+ v i-1ê - v i+1ê x i- x i-1 x i-1ê x i x i+1 x i+ x i+1ê S S 0 S 1 S Figure: Stencil of fifth order WENO scheme
At smooth region, each sub-stencil gives third (k-th) order numerical fluxes v and v +. i+ 1 i 1 For example, on stencil S 1, v (1) i+ 1 v (1)+ i 1 = 1 6 v i 1 + 5 6 v i + 1 3 v i+1 = 1 3 v i 1 + 5 6 v i 1 6 v i+1 Similarly, we have v (0) i+ 1 v () i+ 1 and v (0)+ i 1 and v ()+ i 1 on stencil S 0 on stencil S *The coefficients come from reconstruction process.
Apply a weight to each stencil and a fifth ((k-1)-th) order WENO scheme is obtained. Assume the weights are w 0, w 1, w. Then, we require w r 0, k 1 w s = 1, s=0 for stability and consistency. The fifth order fluxes are given by v i+ 1 k 1 = w r v (r), v + i+ 1 i 1 r=0 k 1 = r=0 w r v (r)+ i 1 Question: How to choose the weights?
If the function v(x) is smooth in all of the candidate stencils, there are constants d r such that v i+ 1 v + i 1 k 1 = d r v (r) i+ 1 = r=0 k 1 r=0 d r v (r)+ i 1 = v(x i+ 1 ) + O( x k 1 ), = v(x i 1 ) + O( x k 1 ). For example, when k=3, d 0 = 3 10, d 1 = 3 5, d = 1 10, d 0 = 1 10, d1 = 3 5, d = 3 10.
In this smooth case, we would like to have Form of the Weights: w r = α r = w r = d r + O( x k 1 ). α r k 1 s=0 α, r = 0,, k 1 s d r (ɛ + β r ) where β r are the so-called smooth indicators of the stencil S r. We require if v(x) is smooth in the stencil S r, then β r = O( x ) if v(x) has a discontinuity inside the stencil S r, then β r = O(1)
Smooth Indicators Let the reconstruction polynomial on the stencil S r be denoted by p r (x). Then, define k 1 β r = l=1 xi+ 1 x i 1 x l 1 ( dl p r (x) dx l ) dx When k=3, β 0 = 13 1 ( v i v i+1 + v i+ ) + 1 4 (3 v i 4 v i+1 + v i+ ), β 1 = 13 1 ( v i 1 v i + v i+1 ) + 1 4 ( v i 1 v i+1 ), β = 13 1 ( v i v i 1 + v i ) + 1 4 ( v i 4 v i 1 + 3 v i )
Lax-Friedrichs Splitting: f ± (u) = 1 (f(u) ± αu), where α = max u f (u) over the relevant range of u. FD WENO Procedure with : Identify v i = f + (u i ) and use WENO procedure to obtain v, and i+ 1 take ˆf + i+ 1 = v i+ 1 Identify v i = f (u i ) and use WENO procedure to obtain v +, and i+ 1 take ˆf i+ 1 = v + i+ 1 Form the numerical flux as ˆf i+ 1 = ˆf + i+ 1 + ˆf i+ 1
Mapped Mapped WENO-Z Scheme g r (w) = w(d r + d r 3d r w + w ) d r + w(1 d r ) α r = g r (w JS w M r = r ) α r s=0 α s Advantage: fifth order accuracy at critical points Disadvantage: the weight of the stencil that contains discontinuity becomes larger after the map; more computational cost.
WENO-Z Scheme Mapped WENO-Z Scheme τ 5 = β 0 β τ 5 αr z = d r (1 + ( β k + ɛ )q ) wr z αr z = s=0 αz s Advantage: fifth order accuracy at critical points with q=; computationally more efficient than mapped WENO.
Bibliography I Mapped WENO-Z Scheme [1] B. Cokcburn, C. Johnson, C.-W. Shu, and E. Tadmor, Advanced numerical approximation of nonlinear hyperbolic equations, Ed. A. Quarteroni, Lecture Notes in Mathematics, vol. 1697, Springer, 1998. [] C.-W. Shu, High order weighted essentially non-oscillatory schemes for convection-dominated problems, SIAM Review, v51 (009), pp.8-16. [3] A. K. Henrick, T. D. Aslam, J. M. Powers, Mapped weighted essentially non-oscillatory scheme: Achieving optimal order near critical points, Journal of Computational Physics 07 (005), pp.54-567.
Bibliography II Mapped WENO-Z Scheme [4] R. Borges, M. Carmona, B. Costa, W. S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, Journal of Computational Physics, 7, (008) pp.3191-311.