Numerical Smoothness and PDE Error Analysis

Transcription

1 Numerical Smoothness and PDE Error Analysis GLSIAM 2013 Central Michigan University Tong Sun Bowling Green State University April 20, 2013

2 Table of contents Why consider Numerical Smoothness? What is Numerical Smoothness? Necessity of Numerical Smoothness The Broader Impact of Numerical Smoothness

3 References T. Sun, Necessity of numerical smoothness, accepted for publication in the International Journal for Information and Systems Sciences, arxiv: v1 [math.na], July T. Sun, Numerical smoothness and error analysis for WENO on nonlinear conservation laws, to appear in Numerical Methods for Partial Differential Equations, T. Sun and D. Rumsey, Numerical smoothness and error analysis for RKDG on the scalar nonlinear conservation laws, Journal of Computational and Applied Mathematics, 241 (2013), T. Sun, Numerical smoothing of Runge-Kutta schemes, Journal of Computational and Applied Mathematics, 233 (2009), T. Sun and D. Fillipova, Long-time error estimation on semi-linear parabolic equations, Journal of Computational and Applied Mathematics, (185) 2006, 1-18.

4 References T. Sun, Necessity of numerical smoothness, accepted for publication in the International Journal for Information and Systems Sciences, arxiv: v1 [math.na], July T. Sun, Numerical smoothness and error analysis for WENO on nonlinear conservation laws, to appear in Numerical Methods for Partial Differential Equations, T. Sun and D. Rumsey, Numerical smoothness and error analysis for RKDG on the scalar nonlinear conservation laws, Journal of Computational and Applied Mathematics, 241 (2013), T. Sun, Numerical smoothing of Runge-Kutta schemes, Journal of Computational and Applied Mathematics, 233 (2009), T. Sun and D. Fillipova, Long-time error estimation on semi-linear parabolic equations, Journal of Computational and Applied Mathematics, (185) 2006, Google NUMERICAL SMOOTHNESS PDE, almost only these.

5 Two fundamentally different error splittings

6 Two fundamentally different error splittings

7 A spatial smoothness indicator Let u R (x) be a piecewise polynomial of degree p or less on a uniform partition a = x 0 < x 1 < < x N = b, h = x i+1 x i = (b a)/n, then its smoothness indicator is S p = ( ˆM, ˆD) with ˆM = ( M 0, M 1,, M N 1 ), where M i = (M 0 i, M1 i,, Mp i ), ˆD = ( D 1,, D N 1 ), where D i = (D 0 i, D1 i,, Dp i ). Furthermore, M k i = d k dx k ur (x + i ), L k i = d k dx k ur (x i ), J k i = M k i L k i For WENO: D k i = J k i /h p+1 k [Sun, 2011] For DG: D k i = J k i /h p+2 k(1+ 1 p ) [Sun, Rumsey, 2010]

8 The error estimate for WENO (smooth solution) Theorem Let u(t, x) be the solution of a nonlinear conservation law u t + f (u) x = 0 and u R n (x) (n = 0, 1,, K) be the reconstructed WENO solution defined in [Sun, 2011]. At the ending time t K = T of the computation, the global error of the numerical solution is u(t K, x) u R K (x) L 1 (Ω) u I u R 0 L 1 (Ω) K 1 +h 5 n=0 τ [ C E (Sn) 5 + C H (Sn) 5 ] K 1 + τ k n=0 τc F (S 5 n, T k n ), where u I is the initial value, u0 R is the numerical initial value, the functions C E (Sn), 5 C H (Sn) 5 and C F (Sn, 5 Tn k ) of the smoothness indicators Sn 5 and Tn k are computable.

9 Definition of Numerical Smoothness A piecewise polynomial u R is numerically C p+1 -smooth in the partition of cell size h, if there is a constant M such that D k i M for all k and i..

10 Definition of Numerical Smoothness A piecewise polynomial u R is numerically C p+1 -smooth in the partition of cell size h, if there is a constant M such that D k i M for all k and i.. u R is numerically H p+1 -smooth in the partition if N 1 i=1 h [ (D 0 i ) 2 + (D 1 i ) (D p i )2 ] M.

11 Definition of Numerical Smoothness A piecewise polynomial u R is numerically C p+1 -smooth in the partition of cell size h, if there is a constant M such that D k i M for all k and i.. u R is numerically H p+1 -smooth in the partition if N 1 i=1 h [ (D 0 i ) 2 + (D 1 i ) (D p i )2 ] M. u R is numerically W p+1,1 -smooth in the partition if N 1 i=1 h [ D 0 i + D 1 i + + D p i ] M.

12 Definition of Numerical Smoothness A piecewise polynomial u R is numerically C p+1 -smooth in the partition of cell size h, if there is a constant M such that D k i M for all k and i.. u R is numerically H p+1 -smooth in the partition if N 1 i=1 h [ (D 0 i ) 2 + (D 1 i ) (D p i )2 ] M. u R is numerically W p+1,1 -smooth in the partition if N 1 i=1 h [ D 0 i + D 1 i + + D p i ] M. Remark: D k i O(1) is equivalent to J k i O(h p+1 k ).

13 The positive definite quadratic form Q Q(D 0, D 1,, D p ) = min ˆv + 1 ˆv P 2 p k=0 k! τ k D k 2 L 2 ( 1 2,0) + ˆv 1 2 p k=0 k! τ k D k 2 L 2 (0, 1 2 ) Lemma Q(D 0, D 1,, D p ) is a positive definite quadratic form.

14 Error estimates from below Theorem Suppose that u H p+1 (a, b), and u R is as above. Then, there is a constants C 2 > 0, independent of h, u and u R, such that u u R L 2 (a,b) h p+1 h Q(Di 0, D1 i,, Dp i ) C 2 u H p+1 (a,b). 0<i<N

15 Error estimates from below Theorem Suppose that u H p+1 (a, b), and u R is as above. Then, there is a constants C 2 > 0, independent of h, u and u R, such that u u R L 2 (a,b) h p+1 h Q(Di 0, D1 i,, Dp i ) C 2 u H p+1 (a,b). 0<i<N Moreover, if u C p+1 [a, b], there is a constants C > 0, such that [ ] u u R L (a,b) h p+1 max Q(D 0 0<i<N i, D1 i,, Dp i ) C u C p+1 [a,b].

16 Error estimates from below Theorem Suppose that u H p+1 (a, b), and u R is as above. Then, there is a constants C 2 > 0, independent of h, u and u R, such that u u R L 2 (a,b) h p+1 h Q(Di 0, D1 i,, Dp i ) C 2 u H p+1 (a,b). 0<i<N Moreover, if u C p+1 [a, b], there is a constants C > 0, such that [ ] u u R L (a,b) h p+1 max Q(D 0 0<i<N i, D1 i,, Dp i ) C u C p+1 [a,b]. If u W p+1,1 (a, b), there is a constants C 1 > 0, such that [ ] u u R L 1 (a,b) h p+1 h Q(Di 0, D1 i,, Dp i ) C 1 u W p+1,1 (a,b). 0<i<N

17 Understanding the error estimates from below Remark: According to the theorem, in order to have it is necessary that N 1 i=1 Moreover, in order to have it is necessary that u u R L 2 (a,b) O(h p+1 ), h Q(D 0 i, D 1 i,, D p i ) O(1). u u R L (a,b) O(h p+1 ), D k i O(1) for all k and i.

18 The Broader Impact of Numerical Smoothness It is reasonable to expect all Di k to be bounded. If they are bounded, we can use them to obtain the optimal-optimal error estimates as in the case of WENO. If they are not, the scheme must have lost optimal convergence rate.

19 The Broader Impact of Numerical Smoothness It is reasonable to expect all Di k to be bounded. If they are bounded, we can use them to obtain the optimal-optimal error estimates as in the case of WENO. If they are not, the scheme must have lost optimal convergence rate. Numerical smoothness gives us a convenient way to find bad schemes. A scheme for a time-dependent problem may very well be Lax-stable, dissipative and/or TVD, but a bad scheme because it does not maintain numerical smoothness, and consequently loses optimal convergence rate.

20 The Broader Impact of Numerical Smoothness It is reasonable to expect all Di k to be bounded. If they are bounded, we can use them to obtain the optimal-optimal error estimates as in the case of WENO. If they are not, the scheme must have lost optimal convergence rate. Numerical smoothness gives us a convenient way to find bad schemes. A scheme for a time-dependent problem may very well be Lax-stable, dissipative and/or TVD, but a bad scheme because it does not maintain numerical smoothness, and consequently loses optimal convergence rate. It has been believed that NUMERICAL STABILITY is what we need for numerical schemes, but for nonlinear equations and complex schemes, error analysis has been impossible. Why? It should have been NUMERICAL SMOOTHNESS to begin with (so far, just personal opinion).

21 The Broader Impact of Numerical Smoothness It is reasonable to expect all Di k to be bounded. If they are bounded, we can use them to obtain the optimal-optimal error estimates as in the case of WENO. If they are not, the scheme must have lost optimal convergence rate. Numerical smoothness gives us a convenient way to find bad schemes. A scheme for a time-dependent problem may very well be Lax-stable, dissipative and/or TVD, but a bad scheme because it does not maintain numerical smoothness, and consequently loses optimal convergence rate. It has been believed that NUMERICAL STABILITY is what we need for numerical schemes, but for nonlinear equations and complex schemes, error analysis has been impossible. Why? It should have been NUMERICAL SMOOTHNESS to begin with (so far, just personal opinion). A lot of work to be done (everybody s opportunity).

22 THANK YOU!