Numerical issues in the implementation of high order polynomial multidomain penalty spectral Galerkin methods for hyperbolic conservation laws

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1 Nuerical issues in the ipleentation of high order polynoial ultidoain penalty spectral Galerkin ethods for hyperbolic conservation laws Sigal Gottlieb 1 and Jae-Hun Jung 1, 1 Departent of Matheatics, University of Massachusetts at Dartouth North Dartouth, MA 747-3, U.S.A. Abstract. In this paper, we consider high order ulti-doain penalty spectral Galerkin ethods for the approxiation of hyperbolic conservation laws. This forulation has a penalty paraeter which can vary in space and tie, allowing for flexibility in the penalty forulation. This flexibility is particularly adventageous for probles with an inhoogeneous esh. We show that the discontinuous Galerkin ethod is equivalent to the ultidoain spectral penalty Galerkin ethod with a particular penalty ter. The penalty paraeter has an effect on both the accuracy and stability of the ethod. We exaine the nuerical issues which arise in the ipleentation of high order ulti-doain penalty spectral Galerkin ethods. The coefficient truncation ethod is proposed to prevent the rapid error growth due to round-off errors when high order polynoials are used. Finally, we show that an inconsistent evaluation of the integrals in the penalty ethod ay lead to growth of errors. Nuerical exaples for linear and nonlinear probles are presented. Key words: High Order Polynoial Galerkin Methods, Penalty Boundary Conditions, Discontinuous Galerkin Methods, Hyperbolic Conservation Laws, Round-off Errors, Truncation Methods. 1 Introduction Polynoial Galerkin ethods are widely used for the nuerical solution of hyperbolic conservation laws [1,, 9, 1]. These ethods seek a polynoial approxiation of the solution for which the projected residual of the differential equation to the polynoial space vanishes. Two such classes of ethods are the spectral Galerkin ethods (sgm) and the discontinuous Galerkin ethods (dgm). Traditionally, sgm have used high order polynoials on one eleent, while dgm use lower order polynoials on any eleents. However, ulti-doain sgm exist and are known as spectral eleent ethods. In order to increase the accuracy of the approxiation, these ethods can use ore saller eleents Corresponding author. Gottlieb) Eail addresses: jjung@uassd.edu (J.-H. Jung), sgottlieb@uassd.edu (S. Global Science Preprint

2 (h-refineent) or raise the degree of the polynoial in each eleent (p refineent). High order polynoials have nuerical issues such as sensitivity to roundoff errors, so it is iportant to carefully study their effects on accuracy and stability when used in ultidoain penalty spectral Galerkin ethods. The penalty forulation penalizes the boundary or interface conditions at each eleent by introducing a penalty ter which includes a penalty paraeter. This penalty paraeter confers a great deal of flexibility on the proble, as it can change over space and tie. We deonstrate the advantages of the flexibility in the choice of penalty paraeter, especially in the case where an inhoogeneous grid syste is used. An inhoogeneous grid can be due to a difference in eleent size or in polynoial order at each eleent, but this type of grid is subject to non-physical reflecting or dispersive odes which ay appear in the solution. By odifying the penalty conditions near the grid discontinuity, we show that the sgm can reduce the non-physical odes while coputing the other eleents efficiently and accurately. We further consider the effects of the penalty ethod on the stability and accuracy. In this context, we show that the dg forulation is a special case of the penalty ulti-doain sgm. Next we discuss the effect of round-off errors for high order sgms. These round-off error effects can arise fro the the ill-conditioned ass atrix for high order polynoials, and the nuerically inconsistent evaluations of the ass atrix and the load vector. The coefficient truncation ethod is introduced to reduce round-off errors. This ethod truncates high order coefficients which do not show rapid decay in the solution of the linear syste, and prevents error growth.the second round-off error effect we explore is the error resulting fro inconsistent evaluations of the two sides of the equation. While the right-hand-side of the linear syste is evaluated by quadrature, the left-hand side can usually be coputed exactly. In the case of orthogonal polynoials, this atrix is a diagonal atrix which siplifies the process of solving the syste. However, coputing one side of the equation exactly and the other by quadrature results in inconsistency errors which rise when the polynoial order is raised. We show these nuerical inconsistency errors, and their dependence on the penalty paraeter, in nuerical coputations. The paper is structured as follows. In Section, we forulate the ultidoain penalty sgm and deonstrate the effect of the penalty ters on the stability and accuracy. The penalty sgm with non-hoogeneous grid and the flexibility of the penalty ethods are discussed. The equivalence of the dgm to the sgm is shown as well. In Section 3, the effect of round-off errors on the high order sgm is discussed. The coefficient truncation ethod is introduced and nuerical consistency is studied for the reduction of round-off errors. In Section 4, we suarize our results and discuss future research directions. Penalty Spectral Galerkin ethod In this paper, we consider the one-diensional hyperbolic conservation law u t +f(u) x =, x Ω=[ 1,1], u:[ 1,1] R + R,t>, (.1)

3 3 with the initial condition u(x,)=g(x), and boundary conditions B ± u(x,t)=h ± (t), x Ω, where B ± are the boundary operators at the doain boundary Ω. For siplicity, we will consider ainly the case f (u), for which the boundary condition is u( 1,t)=h + (t). In practice, this case can be easily generalized using the flux splitting and treating each flux with the appropriate boundary condition. In 1988, Gottlieb and Funaro introduced a penalty boundary condition for the spectral collocation approxiation of Eq. (.1) [4] and there have has been uch research on the penalty collocation ethods [3,5 8]. The ain otivation of penalty boundary conditions for collocation ethods is that the differential equation is satisfied exactly at a given set of collocation points, while at the boundaries we add a ter which penalizes for the approxiation s distance fro the prescribed boundary conditions to ake the given equations satisfied asyptotically at any points. The nuerical approxiation is the polynoial U(x,t)= k= b k(t)p k (x), where P k (x) are the basis polynoials of degree k in x and b k (t) are the unknown expansion coefficients, which will be deterined. The spectral collocation penalty ethod leads to the requireent that U t +f(u) x =τq + (x) ( f (U( 1,t)) f ( h + (t) )), at each of the collocation points x=x j. The penalty paraeter τ can depend on x and t. Q + (x) is a polynoial of degree which vanishes at all the collocation points x j, except at the boundary point x = 1, so that the penalty ter is only applied to the boundary point. To forulate the penalty Galerkin spectral ethod, we assue a solution of the for U(x,t)= k= b k(t)p k (x) and find the expansion coefficients by requiring that the projection of the residual onto the solution subspace vanish. To satisfy the boundary conditions, we can ipose a penalty ter on the strong forulation (as in the collocation case), and then require the projection of the residual of the penalized equation to vanish. Alternatively, we can ipose the penalty ter after the projection. If we choose P k (x) tobea set of orthogonal polynoials, such that Ω P k(x)p j (x)dx = γ k δ jk, the penalty Galerkin forulation yields the syste γ j b j(t)+ x P j (x)dx=τ Ωf(U) j ( f(u( 1,t)) f(h + (t)) ) Q + (x)p j (x)dx Ω for j =,,, where the superscript denotes the derivative with respect to tie and γ j are the noralization factors. We have flexibility in the choice of the penalty paraeter τ j and Q +, provided only that the polynoial Q + ( 1) = 1. One way to accoplish this is to set Q+ (x) tobe polynoials of degree, that is, Q + (x)=1=p (x). In this case, the penalty ters appear

4 4 only in the equation for b (t), that is, γ b (t) = f(u) x dx+τ ( f(u( 1,t)) f(h + (t)) ) Ω = f(u( 1,t)) f(u(1,t))+τ ( f(u( 1,t)) f(h + (t)) ), γ j b j (t) = f(u) x P j (x)dx, j =1,,. Ω We refer to this forulation as the strong forulation. However, this choice of Q + (x)=1 does not lead to good stability properties for a linear hyperbolic wave equation, as we will discuss below. A different choice of Q + (x) ay resolve this proble. For exaple, if we let Q + 1 = P k (x), γ k then the integral in the penalty ter can be evaluated exactly 1 1 Q + 1 (x)p j (x)dx= P k (x)p j (x)dx=p j ( 1). γ l 1 k= 1 k= For the orthogonal polynoials such as Chebyshev or Legendre polynoials, P j ( 1) = ( 1) j. In this case the forulation becoes siilar to the collocation case above: γ j b j(t)= f(u) x P j (x)dx+τ ( f(u( 1,t)) f(h + (t)) ) ( 1) j, Ω for j =,,. We refer to this forulation as the weak forulation. For the linear hyperbolic wave equation this penalty ter yields a better stability profile. To see the difference between these two forulations, consider the siple advection equation u t +u x = with both the strong and weak penalty forulations. Figure 1 shows that the strong forulation (represented by ) has positive real eigenvalues which will lead to instability, while the eigenvalues for the weak forulation (represented by ) are all in the left half plane. This is a clear indication that the choice of the penalty polynoial, Q + (x) is critical for stability. Fro now on, we will consider only the weak forulation polynoial. The penalty paraeter τ also plays an iportant role in the stability of the ethod. Figure shows the eigenvalues with the dgm and the penalty sgm for two different values of τ. The top figures show the eigenvalues in the coplex plane for τ = 1(left) and τ = (right) for polynoial order = 8. In each figure, the sybols and represent the eigenvalues with the dgm and the sgm respectively. Fro the left top figure it is clear that the dgm and sgm are identical for τ = 1, but when τ is increased, as in the top right figure, the real part of one eigenvalue increases. This one eigenvalue alone is responsible for the growth in the spectral radius ρ. This eigenvalue will later be iplicated in penalty sgm s increased sensitivity to roundoff errors for large τ.

5 I(λ) I(λ) Re(λ) Re(λ) Figure 1: Eigenvalues in coplex plane with the strong penalty forulation with =8(left) and = (right) for τ = 1. The sybols and represent the eigenvalues with the weak and strong penalty ethods respectively. The left botto figure in Figure shows the spectral radius ρ increasing as a function of the polynoial order for τ = 1. The right botto figure shows the spectral radius ρ as a function of the penalty paraeter τ, for various polynoial orders, on a logarithic scale. The figure shows that ρ increases fast around τ = 1. The figures show that the stability of the forulation depends on the penalty paraeters τ j, as well as Q + (x). Later we will see that the choice of penalty paraeter will affect the accuracy as well as stability of the ethod..1 Multidoain spectral penalty Galerkin ethods To increase the order of accuracy in the penalty sgms, we can allow the polynoial order to rise or we can divide the doain into any saller subdoains, and apply the sgm in each doain. To forulate the ulti-doain ethod, we divide the doain Ω=[ 1,1] into N subdoains, or eleents, I l,l=1,...,n. For siplicity, we assue here that each eleent has the sae polynoial order,, but this is not necessary, or in fact advisable, in general. The solution in each doain is given by U l (x,t)= b l k (t)p k(ξ(x)), x I l = [ ] x l 1/,x l+1/, k= where I l is the lth eleent with the doain interval x l and x l 1/ = x l x l, x l+1/ = x l + x l where x l the cell center. Finally, ξ(x) is the linear ap ξ : x [ 1,1]. For each eleent, we have for each j =,,N k= db l k (t) dt I l P j (ξ(x))p k (ξ(x))dx+ I l P j (ξ(x))f(u l ) x dx=p j I l, (.)

6 I(λ) I(λ) Re(λ) Re(λ) = 4 = 8 = 1 = 16 Log 1 ρ Log 1 ρ τ Figure : Top: Eigenvalues in coplex plane with τ = 1(left) and τ = (right) for =8. The sybols and represent the eigenvalues with the dgm and the sgm respectively. Botto: The spectral radius ρ versus τ with the dgm(left) and the sgm(right) in logarithic scale. where P j I l is the penalty ter for the jth coefficient. Assuing, as above, that f (U), the penalty ter can take the for P j I = τ j ( f(u ( 1,t)) f(h + (t)) ) ( P j ( 1) ) P j I l = τ j f(u l (x l 1,t)) f(u l 1 (x l 1,t)) P j ( 1) l =1,...,N where the τ j s ay be different in each subdoain. This forulation easily generalizes to the case where f (U) is allowed to be negative. In that case, we split the flux f =f + +f into its positive ( df + df du ) and negative ( du ) parts. For scalar hyperbolic equations, f =f + when f U > and f =f when f U <. For the syste of Eq. (.1), f ± are defined as f ± = A ± du, where A is the Jacobian atrix, i.e., A= f U. The Jacobian A is then given by A ± =T Λ ± T 1,

7 7 where T is the siilarity transforation of A and Λ + and Λ are the atrices coposed of the positive and negative eigenvalues respectively with Λ=Λ + +Λ so that f =f + +f. In this case we can extend the penalty ters to include ters not seen in a characteristic decoposition, so that the ultidoain spectral Galerkin penalty ethod becoes for each interior doain, ( ) P j I l = τ j 1 f + (U l (x l 1/ )),t)) f + (U l 1 xk (x l 1/ )),t)) Q + (ξ)p j (ξ)dξ+ Ω ( ) τ j f (U l (x l 1/ )),t)) f (U l 1 xk (x l 1/ )),t)) Q + (ξ)p j (ξ)dξ+ Ω ( ) τ j 3 f + (U l (x l+1/ )),t)) f + (U l+1 xk (x l+1/ )),t)) Q (ξ)p j (ξ)dξ+ Ω ( ) τ j 4 f (U l (x l+1/ )),t)) f (U l+1 xk (x l+1/ )),t)) Q (ξ)p j (ξ)dξ, (.3) where τ j are the penalty paraeters, and the polynoials Q can be chosen as before. If τ j ==τ j 3, the above penalty forulation is basically the characteristic decoposition. The case where τ j 1 =τ j =τ j 3 =τ j 4 is equivalent to no flux-splitting. Adjusting the coefficients allows for flexibility in applying this ethod. The ajor advantage of the flexibility of this forulation is in the case of the non-hoogeneous grid, as we see in the following exaple. Exaple.1: Consider the following equation q t +f x =, (.4) where q =(u,v) T, and f =(v,u). The initial conditions are u(x,)=sin(ωπx) and v(x,)= with ω = 5. With these initial conditions, the exact solutions are given by u(x,t)= 1 ( sin(ωπ(x t)) sin(ωπ(x+t))) and v(x,t)= 1 ( sin(ωπ(x t))+sin(ωπ(x+t))). This equation was previously investigated by Hu and Atkins for the dgm in [11]. The positive and negative fluxes are given by f + =(u+v,u+v) T and f =(u v,u v) T. We can easily show that the sgm is equivalent to the dgm with the characteristic decoposition if the following penalty paraeters are taken, that is, for q 1 =u, τ 1 = 1 =τ 4, and τ ==τ 3, Ω and for q =v, τ 1 = 1 = τ 4, and τ ==τ 3. On the other hand, the case τ 1 =τ = 1 and τ 3=τ 4 = 1 are taken for q 1=u and τ 1 =τ = 1 and τ 3 =τ 4 = 1 for q =v, there is no flux-splitting. In our nuerical exaple, we consider the sgm approxiation with a non-hoogeneous grid. A total of 5 eleents are used. In the interval x =[ 1,], there are 47 eleents

8 8 u(x,t) x u(x,t) x Figure 3: Solution u(x,t) to Eq. (.4) at t =.15 with =5 with totally 5 eleents. Left: Inhoogeneous grid with no splitting penalty ethod. Right: Inhoogeneous grid with no splitting penalty ethod except 3 interface eleents near x=. The interface of two inhoogeneous edia is at x=. each of size x i = 47 1, i =1,,47. In the interval x =[,1], there are 3 eleents with of size x i = 1 3, i=48,49,5. In each eleent, polynoial order =5 is used. In each interval x=[ 1,] and x=[,1], the solutions are sooth and there is no need to use flux splitting at all. In that case, we use the penalty ethod such that there is no flux-splitting, i.e. τ 1 = τ = τ 3 = τ 4 = 1 for u and τ 1 = τ = 1 = τ 3 = τ 4 for v. If there is no flux splitting, the ethod is easily ipleented without coputing the local flux conditions for the characteristic decoposition. However, the inhoogeneity of the grid raises the issue of possible non-physical reflection waves at the grid discontinuity at x =. The penalty ethod can be easily ipleented to reduce the artificial reflections at the grid discontinuity x =. By adopting the characteristic flux splitting only for the doains near the grid discontinuity, one can reduce the agnitude of the non-physical reflections significantly. Furtherore, one can take an advantage of the no flux splitting ethod inside each hoogeneous doain. The left figure of Figure 3 shows the non-splitting penalty ethod at t =.15. As expected, there are sall fluctuations in the solution in the region x. These fluctuations are the non-physical reflecting solutions reflected at x= and propagating to the left. The right figure shows the result with the penalty ethod where only three eleents near x= and two boundary eleents, use the characteristic penalty ethod while the other eleents use the non-splitting ethod. For the nuerical experient, for exaple, I 46,I 47 and I 48 are ipleented based on the characteristic penalty ethod. Here note that the grid discontinuity x= exists between I 47 and I 47. We also note that we use the characteristic penalty ethod for I 1 and I 5 to iniize the boundary effects. As shown in the figures, the penalty ethod is efficiently flexible to adopt the grid inhoogeneity and obtains an accurate result without any considerable non-physical reflecting odes. In [3] the sae issue, but in the collocation ethod, was also briefly discussed.

9 9. The dgm as a special case of the the ulti-doain penalty sgm In this section, we will present the dgm and then show that it can be seen as an exaple of the ulti-doain penalty sgm with a particular penalty ter τ = 1. To forulate the dgm we begin with the Galerkin for for every eleent I l, db l k (t) P j (ξ(x))p k (ξ(x))dx+ P j (ξ(x))f(u l ) x dx=, j=,,. dt I l I l k= and integrate by parts to obtain, k= db l k (t) P j (ξ(x))p k (ξ(x))dx dt I l dp j (ξ(x)) I l dx f(u l )dx= P j (ξ(x))f(u l ) x l+ 1 x l 1. (.5) for j =,,. As usual, we assue that f (U) without loss of generality. Under this assuption, the boundary ter in the above equation can be evaluated by replacing the left flux with the incoing flux based on the characteristic direction for l>, B j I l := P j (ξ(x))f(u l (x,t)) Il =P j ( 1)f(U l 1 (x l 1,t)) P j (1)f(U l (x l+ 1,t)). In the case l = where the left-ost interval in considered, we replace the incoing value by the given boundary condition, B j I = P j ( 1)f(h + (t)) P j (1)f(U l (x l+ 1,t)). (.6) To see the relationship between the dgm and sgm, we define the auxiliary integrals F and G F = and we note that F +G = I l dp j (ξ(x))) I l dx d ( ) P j (ξ(x))f(u l (x,t)) dx f(u l )dx G = P j (ξ(x))f(u l ) x dx, I l dx=p j (1)f(U l (x l+ 1,t)) P j ( 1)f(U l (x l 1,t)). (.7) With these two integrals and the penalty and boundary ters, the dgm and sgm can be rewritten, for each doain l: (dg) M b F =B Il, (sg) M b +G =P Il, (.8) where the ass atrix M and the coefficient vector b are defined by ( T M jk = P j (ξ(x))p k (ξ(x))dx, b= b l (t),,bn(t)) l. I l

10 1 x 1 7 x Figure 4: Exaple.: On the horizonal axis is the value of τ while on the vertical axis the L error of the nuerical solution at tie t =.1, with N =1. Left: The errors as a function of τ for =3. Right: The errors as a function of τ for =6. To show that the dgm and sgm forulations are equivalent, we can rearrange the dg forulation Eq. (.8) for l> M b =F +B Il =P j (1)f(UN l (x l+ 1,t)) P j ( 1)f(UN l (x l 1,t)) G+B Il, M b +G = P j (1)f(U l (x l+ 1,t)) P j ( 1)f(U l (x l 1,t))+B Il = P j (1)f(U l (x l+ 1,t)) P j ( 1)f(U l (x l 1,t)) +P j ( 1)f(U l 1 (x l 1,t)) P j (1)f(U l (x l+ 1,t)) ( = P j ( 1) f(u l 1 (x l 1,t)) f(u l (x l 1,t)) which is equal to the penalty ter P Il with τ = 1. For the case l=, the left boundary ter U l 1 (x l 1,t) is replaced by the boundary condition h + (t). Thus, the dgm forulation is just a special case of the penalty ultidoain sgm. We saw in Figures 1 and that different values of the penalty paraeter yield different stability properties. Additionally, changing the value of τ will also change the size of the errors. To see this, consider the following exaple. Exaple.: Consider the sgm for u t +u x = with the initial condition u(x,)= sin(πx) and periodic boundary conditions. Using the ultidoain sgm with N =1 and the weak penalty forulation, we exaine the the effect of the penalty paraeter τ on the errors. Figure 4 shows the L errors for different τ and =3(left) and =6(right) at t=.1. The figures show that the optial value of τ is not τ = 1, i.e. the dgm is not the optial ethod within the class of sgms. Furtherore, we observe that the optial value of τ also depends on the polynoial order. Note that L errors are 1 and 1 9 for =3 and =6, respectively. )

11 11.3 The flexibility of the penalty paraeter One of the ain advantages of the penalty sgm is that the penalty paraeters can vary depending on the proble. To see how the penalty paraeter affects the peforance of the evaluation of hyperbolic conservation laws, consider the following nuerical exaple. Exaple.3: Consider Burgers equation u t +(u /) x =, x [ 1,1], t>, (.9) with the initial condition u(x,)=x. Then the exact solution u e (x,t) is given by u e (x,t)= x 1+t. With the given initial condition, there is no incoing boundary condition for the boundaries both at x= 1 and x=1. Also, if x>, the characteristic incoing boundary conditions are applied at the left boundary of each eleent, I l.ifx<, the characteristic incoing boundary conditions are applied at the right boundary of each eleent, I + l. Since the solution is a polynoial of degree only 1, the approxiation in each eleent can be sufficiently resolved with =1, i.e. U = 1 k= b k(t)p k (x). However, the expansion coefficients b k, for k>1 are not necessarily zero in the Galerkin approxiation. The weak sgm iposes the interface conditions for every ode for which the overall approxiation is affected by round-off errors when the high order approxiation is sought. The penalty sg forulation has ore flexibility than the dg forulation to deal with such issue by exploiting the penalty paraeters. Since the solution is only a polynoial of degree 1, we eploy the following penalty paraeters { τ j τ if j =,1 = otherwise. (.1) This procedure can be autoated by exaining the regularity of the coefficients, and then designing penalty paraeters that take this into account. Figure 5 shows the sg forulation with τ j = 1, j =,, which is equivalent to the dgm and the odified penalty forulation with the condition of Eq. (.1). The figures show the L and L errors at t =.15. We use the Legendre polynoials with the total nuber of eleents being 3, with =, τ = 5 and CFL=.1. As shown in the figure, the odified penalty ethod represented by yields the best results. We also copute the errors between the exact expansion coefficients and the approxiated coefficients. The exact expansion coefficients for each doain are given in Table 1 at any tie t. Figure 6 shows the errors in the expansion coefficients for both the dgm approxiations and the odified penalty approxiations at t=t f =.15. The figures show that the errors of each expansion coefficient are larger than 1 14 for the dgm approxiation for b k,k>1 while they are close or below 1 14 for the odified penalty sgm.

12 1 SGM dgm SGM dgm Log 1 L Log 1 L Figure 5: Exaple.3: The L (left) and L (right) errors versus for Burgers equation. N =3, =, τ = 5, CFL=.1, and t f =.15. The sybols and represent the dgm and the odified sgm. Eleent 1 Eleent Eleent 3 Eleent 1 Eleent Eleent 3 Log 1 b b exact Log 1 b b exact Figure 6: The errors between the exact and approxiated coefficients in Exaple.3. Left: dgm approxiation. Right: Modified penalty approxiation. The eleent 1 is in x [ 1, 1 3 ], the eleent in x [ 1 3, 1 3 ] and the eleent 3 in x [ 1,1]. 3 Table 1: The exact expansion coefficients for Exaple.3, for each of three doains at tie t. x [ 1, 1 3 ] x [ 1 3, 1 3 ] x [1 3,1] b (t) t 3 1+t b 1 (t) 3 1+t 3 1+t 3 1+t b i (t),i>1 3 Roundoff errors of high order sgm 3.1 The coefficient truncation ethod Although a larger polynoial order should produce a ore accurate solution, in practice we see that Galerkin approxiations with higher order polynoials are ore sensitive to

13 13 roundoff errors. This sensitivity to roundoff errors can destroy the accuracy of a solution for large polynoial values. In this section, we present nuerical exaples of this proble, and suggest the coefficient truncation ethod to resolve it. Exaple 3.1: Consider, once again, the linear advection equation U t +U x =, x [ 1,1], t>. The solution of this equation is approxiated using a dg forulation with the onoial basis function ψ i (x)=x i. Figure 7 shows the decays of the transfored vector b (left) and the obtained expansion coefficient vector x (right) for the last eleent which contains the the right boundary x = 1. The paraeters used for the nuerical approxiation are total nuber of doain = 5, =37, t f =.11, and CFL=.5. The left figure in Figure 7 shows that the eleent of b decays with until around 33 with the agnitude 1. Beyond >33 it is observed not to decrease. This is due to the large condition nuber, κ, of the stiffness atrix M, i.e. κ The right figure in Figure 7 shows the decay or growth of the evaluated expansion coefficient vector x. The figure clearly shows that the expansion coefficients grow rapidly with. Figure 8 deonstrates that this results in large errors. This siple exaple shows that for high order polynoials, the calculation of the coefficients is very sensitive to round-off error, and this affects the accuracy of the solution. To resolve this proble, we odify the penalty ethod with the truncation ethod introduced in [13] for use in the inverse polynoial reconstruction ethod (IPRM) [1, 14] to reduce deterioration of the error when the polynoial error was large. To apply the coefficient truncation ethod, we look at the coefficients resulting fro the interediate step in the Gaussian eliination. Suppose that the set of expansion coefficients is to be found fro the linear equation Mb =h, then the syste is solved by Gaussian eliination to yield an upper triangular atrix syste U b =c. The coefficients of this syste should be rapidly decaying, so to reduce the proble of round-off errors we ipose this requireent by setting c i = { ci if c i >ɛ t otherwise. (3.1) Here ɛ t is the tolerance level which is to be deterined depending on the decay rate of c. The otivation behind this ethod is that, as shown in Figure 1, the spectral radius ρ increases with for the Galerkin ethod and this akes the ethod sensitive to roundoff errors. The truncation ethod tries to reduce the agnitude of ρ by truncating the RHS of the linear syste of the Galerkin ethod by reducing the rank of the atrix. For exaple, if the truncation order is n, then the rank of the truncated atrix becoes +1 n. With the rank reduced, ρ is also reduced. Exaple 3.: To deonstrate the success of the truncation ethod in reducing the effects of round-off errors, we return to Exaple 3.1 above. We apply the truncation ethod with various tolerence levels, i.e. ɛ t =1 14 (+), ɛ =1 1 ( ), ɛ =1 1 ( ), and ɛ=1 ( ). The left figure in Figure 7 shows that the the eleents of b decay faster when the truncation error is applied for different values of ɛ. The right figure in Figure 7 shows the evaluated expansion coefficient vector x. The figure clearly shows that the expansion

14 14 Log 1 b No Truncation = 1 14 = 1 1 = 1 1 = Log 1 b No Truncation = 1 14 = 1 1 = 1 1 = Figure 7: Exaples 3.1 and 3.: Left: The decay of the eleents of the load vector c with and without the truncation ethod. Right: The decay of the obtained expansion coefficient vector x with and without the truncation ethod. The dgm is used for the advection equation with onoial basis functions. Total nuber of doain is 5, =37, t f =.11, and CFL=.5. Various tolerance levels ɛ t are used, i.e. ɛ t = 1 14,1 1,1 1,1. Log 1 L No Truncation = 1 14 = 1 1 = 1 1 = Log 1 Error 8 1 = 1 1 No Truncation = 1 14 = 1 1 = x Figure 8: Exaples 3.1 and 3.: Left: The L errors versus. Right: The pointwise errors when =37 is used with and without the truncation ethod. The dgm is used for the advection equation with onoial basis functions. Total nuber of doain is 5, t f =.11, and CFL=.5. Various tolerance levels ɛ t are used, i.e. ɛ t =1 14,1 1,1 1,1. coefficients grow with exponentially without the truncation ethod while they decay if the truncation ethod is applied. The figures also shows that the expansion coefficients becoe truncated to alost achine zeros or very close to zeros if the truncation ethod is applied. In the figure, these truncated values are not displayed as the plot uses the logarithic scale. The larger ɛ t is used, the faster the truncation of x occurs. The L errors with (left) and the pointwise errors with =37 (right) are given in Figure 8 with the sae sybols as in Figure 7. The left figure in Figure 8 shows the L decay or growth with with and without the truncation ethod. The figure shows that the L errors decay up to 7 and they start to grow beyond if the truncation ethod is not applied or the truncation ethod is applied with ɛ t =1 14. The truncation ethod with ɛ =1 1 or 1 1 yields the best results up to 4. If ɛ t =1 is used,

15 15 the L reains around 1 for large. It is observed that for >4, the L errors reains around 1 when ɛ t =1 while the other cases result in the growth of L errors. The right figure in Figure 8 shows the pointwise errors with = 37. The figure shows that the best result is obtained when ɛ t =1 1 is used. Figure 8 suggests that the tolerance level used with the truncation ethod can be chosen by observing the decay rate of c for a given. Also, one can use the different tolerance level ɛ t for different eleent since each eleent has different decay rate of c. 3. Nuerical consistency Another factor which increases the sensitivity to round-off errors is the inconsistent evaluation of the integrals in the forulation of the syste of ODEs for the approxiation coefficients. On the Galerkin forulationwe need to copute two sets of integrals, I 1 = Ω P k(x)p j (x)dx and I = Ω f(u) xp j (x)dx. While I 1 can be evaluated exactly, because the for of the polynoials is known explicitly in general, I can not generally be evaluated, and so quadrature ust be used. We define the integral operators E and Q, where E is the exact integration operator E(g)= Ωg(x), and Q soe quadrature rule such as Gauss quadrature rule used to evaluate the given integral Q(g)= M i= g iω i Ωg, where ω i are the weights. The Galerkin forulation can now be perfored in one of four ways: E(P l P j )=E(f(U) x P j )+P l, E(P l P j )=Q(f(U) x P j )+P l, (3.) Q(P l P j )=E(f(U) x P j )+P l, Q(P l P j )=Q(f(U) x P j )+P l. We say that the Galerkin forulation is nuerically consistent if the integrals are evaluated in the sae way on both sides of the equations (both by E or both by Q), and it is nuerically inconsistent if the two integrals are evaluated differently on the two sides of the equation. The phenoenon we are considering is essentially a nuerical one. For high polynoial order, the nuerically consistent forulation yields ore accurate results, and lower sensitivity to roundoff errors, than the inconsistent forulation. Consequently the above 4 different forulations, Eq. (3.), show different errors. Exaple 3.3: We first consider the following steady-state proble u t +u x =sin(πx), x [ 1,1],t>, with initial condition u(x,) = u s (x)+ɛ M δ(x), where ɛ M is achine accuracy, ɛ M =1 16 and δ(x) is the rando function, δ(x) 1 with the noral distribution. The boundary condition is u( 1,t)= 1 π cos( π), t>. The steady-state solution, us =li t u(x,t)= u s (x) for t is then given by u s (x,t)= π 1 cos(π(x t)). For the nuerical experient, the final tie is t f =1, the total nuber of doain N =3, the penalty paraeter τ = 5, and the CFL nuber CFL=.1 for dt=cfl dx= are used where each eleent has the sae eleent size and polynoial order.

16 16 SGM QQ SGM EQ dgm QQ dgm EQ SGM QQ SGM EQ dgm QQ dgm EQ Log 1 L 8 1 Log 1 L Figure 9: Left: L errors vs for the steady-state proble u t+u x = sin(πx). The sybols and denote the quadrature-quadrature (QQ) and exact-quadrature (EQ) forulations for the penalty sgm and and the QQ and EQ forulations for the dgm. The final tie t f =1 and the total nuber of doain is 3. The penalty parater is τ = 5 and the CFL nuber is CFL=.1, i.e. dt= Figure 9 shows the convergence of L and L errors with for the the weak penalty forulation and the dg forulation. The LHS or RHS are evaluated either using the quadrature rules labeled by Q or the exact forula denotedlabeled by E for the penalty forulation with τ = 5. As the figures show, both the L and L errors decay exponentially until around 7. For >7, these errors grow slightly due to round-off errors. The figures show that the results with the consistent evaluation of the stiffness atrix of the sgm (QQ) show better perforance for the weak penalty sgm when round-off error becoe doinant. We note that although the dgm does not suffer fro an inconsistent forulation, the consistent forulation for the penalty sgm yields the best results when is large. 4 Suary and conclusion We describe the forulation of the ulti-doain penalty sgm and deonstrated that the flexibility of the penalty forulation can be advantageous, because it allows us to tailor the penalty paraeter to atch the proble. This is especially relevant in the case where the sub-doains have different esh size or polynoial order, as the flexibility in the penalty paraeters can allow us to avoid costly flux splitting except near the grid discontinuity. However, different values of the penalty paraeter effect the stability, accuracy, and sensitivity to round-off errors. For exaple, the dgm is siply a sgm with a particular choice of penalty paraeter, which for a linear wave equation has nice stability properties, though it is not optial in ters of accuracy. We presented two nuerical issues which arise in the solution of hyperbolic conservation laws using the ulti-doain penalty sgm with high order polynoials. The first is the sensitivity of high order sgms to roundoff errors, which can potentially ruin the accu-

17 17 racy of the solution. To resolve this issue we introduce the coefficient truncation ethod which prevents the rapid growth of the errors with, and has a stabilizing effect on the ethod. The second nuerical issue is that consistent evaluation of the stiffness atrix and the load vector yields a better result when the high order polynoials are used with the penalty forulation. This sensitivity, too, depends on the penalty paraeter, and we note that the case of τ = 1, (the dgm case) does not see affected by this nuerical consistency issue. Future studies will center around ethods for choosing the penalty paraeter to optiize for accuracy and stability, as well as further developent of the coefficient truncation ethod for ulti-diensional probles. 5 Acknowledgeents The authors gratefully acknowledges the advice of David Gottlieb and Jennifer Ryan. This work has been supported by the NSF under Grant No. DMS References [1] B. Cockburn, C. Johnson, C.-W. Shu, E. Tador, Advanced nuerical approxiation of nonlinear hyperbolic equations, Lecture Notes in Matheatics, Springer, Berlin, [] B. Cockburn, G. Karniadakis, C.-W. Shu, The developent of discontinuous Galerkin ethods, in: Discontinuous Galerkin ethods (Newport, RI, 1999), Lecture Notes in Coputer Science and Engineering, vol. 11, Springer, Berlin, pp [3] W.-S. Don, D. Gottlieb, J.-H. Jung, A ultidoain spectral ethod for supersonic reactive flows, J. Coput. Phys. 19 (3) [4] D. Funaro, D.Gottlieb, A New Method of Iposing Boundary Conditions for Hyperbolic Equations, Math. Coput. 51 (1988) [5] D. Funaro, D. Gottlieb, Convergence Results for Pseudospectral Approxiations of Hyperbolic Systes by a Penalty-type Boundary Treatent, Math. Coput. 57 (1991) [6] J. S. Hesthaven, D. Gottlieb, A Stable Penalty Method for the Copressible Navier Stokes Equations: I. Open Boundary Conditions, SIAM Journal on Scientific Coputing 17 (1996) [7] J. S. Hesthaven, A Stable Penalty Method for the Copressible Navier Stokes Equations: II. One-Diensional Doain Decoposition Schees, SIAM Journal on Scientific Coputing, 18 (1997) [8] J. S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral ethods for tie-dependent probles (Cabridge, 7), Cabridge UP. [9] J. S. Hesthaven and T. Warburton,, High-Order Nodal Methods on Unstructured Grids. I. Tie-Doain Solution of Maxwell s Equations, J. Coput. Phys. 181(1), [1] F. X. Giraldo, J. S. Hesthaven, and T. Warburton,, Nodal High-Order Discontinuous Galerkin Method for the Spherical Shallow Water Equations, J. Coput. Phys. 181(), [11] F. Q. Hu, H. L. Atkins, Eigensolution analysis of the discontinuous Galerkin ethod with nonunifor grids: I. one space diension, J. Coput. Phy. 18 ()

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