Hyperbolic Systems of Equations Pose on Erroneous Curve Domains Jan Norström a, Samira Nikkar b a Department of Mathematics, Computational Mathematics, Linköping University, SE-58 83 Linköping, Sween ( jan.norstrom@liu.se). b Department of Mathematics, Computational Mathematics, Linköping University, SE-58 83 Linköping, Sween (samira.nikkar@liu.se). Abstract The effect of an inaccurate geometry escription on the solution accuracy of a hyperbolic problem is iscusse. The inaccurate geometry can for eample come from an imperfect CAD system, a faulty mesh generator, ba measurements or simply a misconception. We show that inaccurate geometry escriptions might lea to the wrong wave spees, a misplacement of the bounary conitions, to the wrong bounary operator an a mismatch of bounary ata. The errors cause by an inaccurate geometry escription may affect the solution more than the accuracy of the specific iscretization techniques use. In etreme cases, the orer of accuracy goes to zero. Numerical eperiments corroborate the theoretical results.. Erroneous computational omain Consier the following hyperbolic system of equations, in two space imensions, W t + ÂW + ˆBW y =, (, y) Ω, t (, T ], LW = g(, y, t), (, y) δω, t (, T ], () W = f (, y), (, y) Ω, t =, in which the solution is represente by the vector W = W(, y, t). Â an ˆB are constant symmetric M M matrices, Ω is the spatial omain with the bounary δω. The bounary operator L is efine on δω, f (, y) R M an g(, y, t) R M are the ata in the problem. Preprint submitte to Journal of Computational Physics January 3, 6
Equation () is transforme to curvilinear coorinates (ξ, η) by (V ξ, V η, V t ) = [J](V, V y, V t ) T, where [J] is the Jacobian matri of the transformation. The transforme problem is JW t + AW ξ + BW η =, (ξ, η) Φ, t (, T ], LW = g(ξ, η, t), (ξ, η), t (, T ], W = f (ξ, η), (ξ, η) Φ, t =, () where A = Jξ Â + Jξ y ˆB, B = Jη Â + Jη y ˆB an J = ξ y η η y ξ > is the eterminant of [J]. The energy metho together with the metric ientities an the use of the Green-Gauss theorem yiels t W(ξ, η, t) J = W T C W s, (3) where the norm is efine by V J = Φ V T J V ξ η. In (3), C = (A, B) n = αa + βb = XΛX T, n = (α, β) is the unit normal vector pointing outwar from Φ, X contains the eigenvectors as columns an Λ contains the eigenvalues on the main iagonal. For more etails see []. The problem () is well-pose if we impose characteristic bounary conitions an the following energy rate is obtaine t W(ξ, η, t) J + W T C + W s = g T Λ g s, (4) where Λ contains the negative eigenvalues of C an C + = XΛ + C T... Errors ue to the wrong position of bounary Consier two hyperbolic problems with solutions U an V pose on two nearby omains. The two omains, epicte in Figure, are both mappe to the unit square. We consier Ω c to be the correct omain an Ω e to be the erroneous one (subscripts c an e enote correct an erroneous, respectively). The transforme equations become J c U t + A c U ξ + B c U η =, J e V t + A e V ξ + B e V η =, (5) where the matrices are A i = [Jξ Â + Jξ y ˆB] i an B i = [Jη Â + Jη y ˆB] i, for i = c,e. Our first result follows immeiately. Since A c A e, B c B e we realize that: ) The wave-spees given by the eigenvalues of the matrices will iffer, an that V will have the wrong wave spees.
y η Ω c Φ Ω e ξ)ξ) Figure : A schematic of two ifferent geometry escriptions both mappe to the unit square. ξ Net, we consier the results of the energy metho given by (4) where W {U,V } an C i = α i A i + β i B i for i = c,e. A closer look at the bounary term C i reveals that the normals as well as the eigenvalues of C i are moifie by the erroneous geometry. This paves the way for the secon, thir an fourth conclusions: ) Error in the normal: This error is cause by imposing the wrong bounary operator ue to an erroneous normal. Consier for eample the soli wall no penetration conition (u,v) n = αu + βv = g = for the Euler equations. An erroneous normal will lea to the wrong bounary operator. This means that the bounary operator L in () is wrong while the ata g is correct. See case in Figure. 3) Error in the position: Here the error is ue to the misplacement of the bounary conition. Consier again the soli wall no penetration conition for the Euler equations, now with a correct bounary operator an ata, impose at the wrong position in space. In this case, the bounary operator L an the ata g in () are both correct. See case in Figure. 4) Error in the ata: Bounary conitions with ata from Ω c but impose at Ω e will also lea to inaccurate results. In this case, the ata g in () is wrong while the bounary operator L can be either correct or wrong. See case an 3 in Figure. We can also have a combination of all the errors. These errors might be, an often are [4], more important than the orer of accuracy of the specific iscretization techniques use (which will be shown later in the numerical eperiments section below). To further iscuss the effect of the wrong position of bounary conitions, consier () an two types of bounary ata, correct an erroneous enote by g c an g e, respectively. The correct an erroneous solutions are enote by U an V, respectively, an for simplicity we assume that they have the same initial 3
(c) = correct geometry (e) = erroneous geometry ne nc ne nc ne 3 nc Possible error sources: normal wrong at right position normal right at wrong position 3 normal wrong at wrong position Figure : A schematic of the erroneous an correct (n c = correct normal, n e = erroneous normal) bounary efinitions. ata f which vanishes close to the bounaries. By subtraction we obtain an error equation with zero initial ata an non-zero bounary ata. Let E = U V enote the error, an δg = g e g c = ( g) e + O( ) = O( ). (6) In (6), = (, y), = c e, y = y c y e, see Figure 3. (c) = correct geometry (e) = erroneous geometry (!, y! )!!!! (!, y! ) Figure 3: A blow-up of two nearby bounary points which result in two types of ata, g c = g( c,y c,t) an g e = g( e,y e,t). The energy metho applie to the transforme version of the error equation gives the corresponing energy rate to (4), as t E J e + E T C e + E s = δg T Λ e δg s, (7) 4
y y in which Λ e contain the negative eigenvalues of C e. From (7) we can conclue that the error is boune for long time calculations, see [3] for etails.. Numerical eperiments To quantify the error stemming from an erroneous geometry we consier the geometries in Figures 4 an 5. The geometry escription is given by.. a a'.8.8.6.4 b.6.4 ' b'.. c c' -. -...4.6.8. Figure 4: The correct omain, Ω c. -. -...4.6.8. Figure 5: The erroneous omain, Ω e. y a () = a sin(ωπ) +, Ω c : b (y) = a sin(ωπy) +, y c () = a sin(ωπ), (y) = a sin(ωπy), y a () = y a () â sin( ˆωπ), Ω e : b (y) = b (y) + â sin( ˆωπy), y c () = y c () â sin( ˆωπ), (y) = (y) + â sin( ˆωπy), (8) where a =., ω = an ˆω =. Moreover, â =. for q = an â = q for q {,, 3}. Note that Ω e in Figure 5 is eaggerate for a better visualization, an Ω c is inclue for clarity. It is not use uring the numerical eperiments. We consier the two-imensional constant coefficient symmetrize Euler equations in which V =[ρ, u, v, T ] T is the perturbe solution. The components ρ, u, v, an T are respectively the ensity, the an y velocity components an the temperature perturbation. The matrices  an ˆB are given in []. To verify the accuracy of our computational metho, we use the manufacture solution V = [sin(θ), cos(θ), sin(φ), cos(φ)] T, where θ(,t) = 5t, φ(y,t) = 5y t. The manufacture solution is injecte in the problem through a forcing function, bounary an initial ata. We use characteristic bounary conitions as in the erivations above. The convergence rates using a finite ifference operator on summation-by-parts form (SBP84) in space an time with weak 5
Convergence Rates, P Energy Norm of the Errors bounary an initial conitions, are shown in Table. SBP84 in space an time converges with 5th orer an the convergence rates are clearly correct []. Table : Convergence rates for the problem () with correct ata Number of gri points 3 4 5 6 7 8 The ρ component 4.498 4.487 4.58 4.654 4.6963 4.7487 The u component 4.6578 4.686 4.6 4.663 4.74 4.777 The v component 4.94 5.875 5.93 4.976 4.8635 4.8394 The T component 5.38 5.349 5.547 4.996 4.884 4.855 Equippe with a 5th orer convergent computational scheme (we use SBP84 in space an time, see [] for etails on this technique), we procee to investigate the problem with erroneous ata. We compute the erroneous solution by computing on the erroneous omain Ω e, taking the bounary ata from the nearby omain Ω c an finally subtracting the eact manufacture solution. The convergence rates for the erroneous solution are shown in Figure 6. We have consiere ifferent magnitues for the eviation of the erroneous omain from the correct omain in terms of = O(h q ) where h is the gri spacing an q {,,, 3}. As seen in Figure 6, imposing the bounary ata at an erroneous position affects the results an reuces the global orer accuracy of the scheme to q. Finally, in Figure 7 we show how the error behaves in long time calculations. It reaches an error boun as preicte above, since we use characteristic bounary conitions as shown in [3]. Convergence Rates for Erroneous Solutions Long Time Calculations 4 3.5 q= q= q= q=3..9 Eact q= q= q=3 3.8.5.7.6.5.5.5.4.3.. -.5 3 4 5 6 7 8 9 N Figure 6: Convergence rates for the erroneous solutions in Ω e with q {,,,3}, T= an sufficiently small time steps. 3 4 5 Time Figure 7: Error versus time for eact an erroneous solutions with q {,,3}, 3 an 5 gri points in space an time, respectively. 6
3. Summary an conclusions The effects of an erroneous geometry escription were iscusse by using the energy metho as the analytical tool. We erive an error estimate which escribes the effects of imposing correct bounary ata at erroneous bounary positions. We use a 5th orer provable stable an convergent high orer finite ifference metho as our computational tool. The results obtaine are vali for all iscretization techniques, that are sufficiently high orer accurate an stable. We conclue that an erroneous geometry escription might lea to the wrong wave spees, a misplacement of the bounary conitions, to the wrong bounary operator an a mismatch of bounary ata. The errors cause by an inaccurate geometry escription may affect the solution more than the accuracy of the specific iscretization techniques use. In etreme cases, the orer of accuracy goes to zero. The numerical eperiments corroborate this conclusion. 4. References [] S. Nikkar an J. Norström, Fully iscrete energy stable high orer finite ifference methos for hyperbolic problems in eforming omains, Journal of Computational Physics, 9():8-98 (5). [] M. Svär an J. Norström, Review of summation-by-parts schemes for initial-bounary-value problems, Journal of Computational Physics, Volume 68, pp. 738 (4). [3] J. Norström, Error boune schemes for time-epenent hyperbolic problems, SIAM Journal of Scinetific Computing, 3:46-59, (7). [4] I. Babuska an J. Chleboun, Effects of uncertanities in the omain on the solution of Neumann bounary value problems in two spatial imensions, Math. Comput., 7:339-37, (). 7