Dispersion and dissipation error in high-order Runge-Kutta ...

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5.2.1 One-dimensional Fourier analysis We are primarily interested in wave propagation and in the associated dispersion and dissipation error of the RKDG scheme. We consider the one-dimensional semi-discrete scheme (16) dq h dt = Aq h in the domain Ω = [−1, 1] filled with vacuum (or air) and assume a monochromatic plane wave (which is also a Fourier mode) q h (0) = q 0 h = [ ] e ikx h , e ikx h T as initial condition. Here, x h represents the vector of the nodes used for the spatial discretisation. We denote the angular wave frequency with ω. The exact wave number k is given by the dispersion relation k 2 = ω 2 /c 2 , with c being the speed of light. The time-exact discrete Fourier mode at time level t n = n∆t will read q h (n∆t) = ν n q h (0) = e −iωn∆t [ e ikx h , e ikx h] T (26) with exact amplification factor ν n = e −iωn∆t and i 2 = −1. To see the effect of the timestepping method, we replace the exact amplification factor ν n with its discrete counterpart νh n and take the fully discrete Fourier mode as q n h = νn h q0 h = νn h [ ] e ikx h , e ikx h T . (27) In addition, we write the SSP-RK scheme as a two-level explicit scheme. Thus q n+1 h = Bq n h (28) holds with amplification matrix B = m∑ l=0 1 l! (∆tA)l (29) and with m being the order of the SSP-RK time-stepping scheme. Substituting (27) into (28) results in the equation [ ] ν n+1 h e ikx h , e ikx h T [ ] = Bν n h e ikx h , e ikx h T , which, after division with νh n , reduce to the eigenvalue problem ν h q 0 h = Bq 0 h. (30) Solving this eigenvalue equation will produce p + 1 different values for ν h,j (and as many corresponding eigenvectors q 0 h,j ). Bearing in mind that ν h,j = e −i˜ω h,j∆t , 12
with complex numerical frequencies ˜ω h,j , we can establish the dispersion and dissipation properties of the scheme. For that, we consider the real (for dispersion) and imaginary (for dissipation) parts of the complex numerical frequencies ˜ω h,j = (i/∆t) ln ν h,j , that is ω h,j = Re [˜ω h,j ] = Re [(i/∆t) ln ν h,j ], ρ h,j = Im [˜ω h,j ] = Im [(i/∆t) ln ν h,j ] , with real numerical frequency ω h,j and numerical dissipation ρ h,j , both corresponding to the eigenvalue ν h,j . One of the computed modes will be close to the frequency of the physical mode. This represents the approximation properties of our scheme. The other modes are spurious. To decide which eigenvalue should be considered, we define the dissipation error as err diss h,j = |ρ h,j | − 1 and the dispersion error as the absolute value of dispersion error err disp h,j = |ω − ω h,j |. The numerical Fourier mode is now taken as the closest eigenvalue to the physical mode { √ } ( ) 2 ( ) ν h := ν h,j : min err disp j h,j + err diss 2 h,j . (31) In the numerical dispersion and dissipation analysis of the fully discrete schemes, we consider a wide range of values for ∆t and h k . The eigenvalues of (30) are computed in Matlab. It is also important to consider the convergence of the numerical dispersion and dissipation error. In [1] a complete dispersion and dissipation analysis of the semi-discrete advection equation was carried out for discontinuous Galerkin methods with high-order tensor-product elements. In that article it was proven that in the asymptotic region hk = 2πh → 0 (with λ being the wave length) the dispersion relation for a pth-order λ method is accurate to order 2p + 3 for the dispersion error and order 2p + 2 for the dissipation error. See [22] and [1] for more details. In a more recent work [2] the dispersion and dissipation analysis of the semi-discrete wave equation was provided for some loworder schemes (linear elements for the general DG and up to third order elements for the IP-DG) and the authors also conjectured on how the results would extend to arbitrary order elements. For the fully discrete system we consider here, the rate of convergence is also influenced by the time-stepping method. However, for most polynomial orders p, the convergence of the dispersion and dissipation error still by far supersedes that of the error measured in the l 2 -norm. The numerical results are discussed in Section 6. 5.2.2 Two-dimensional Fourier analysis As in one dimension, we assemble our right-hand side into a global matrix and consider the abstract Cauchy problem (16) dq h dt = Aq h where now q = [H x h , Hy h , Ez h ]T and the matrix A represents the semi-discrete system resulting from the discretisation of (25). The only difference in the mathematical formulation to the one-dimensional case is that the dispersion relation now reads k 2 x + k 2 y = ω 2 /c 2 . The time-exact discrete Fourier mode satisfying (25) is equal to q n h = νn [ (k y /ω)e ikxx h+ik yy h , (−k x /ω)e ikxx h+ik yy h , e ikxx h+ik yy h ] T (32) 13
Page 1 and 2: Dispersion and dissipation error in
Page 3 and 4: ehaviour of the DG method for the t
Page 5 and 6: 3 Discontinuous Galerkin discretisa
Page 7 and 8: The right-hand side of (10) is resp
Page 9 and 10: 4 Runge-Kutta time-stepping methods
Page 11: or they can be expressed by the wav
Page 15 and 16: Perhaps even more illuminating is t
Page 17 and 18: that the convergence rate of the di
Page 19 and 20: [22] F. Q. Hu and H. L. Atkins. Eig
Page 21 and 22: p=2 p=4 1 0.8 0.6 0.4 0.2 0 −0.2
Page 23 and 24: 40 1 35 30 DoF / λ 25 20 0.05 15 0
Page 25 and 26: 0.2 0.0001 3e−05 1e−05 ∆ t /
Page 27 and 28: DoF / λ DoF / λ ∆ t / ∆ t max
Page 29 and 30: 15 Frequency error is 0.001 15 Freq
Page 31 and 32: Table II: Convergence of the global
Page 33 and 34: Table IV: Computational work needed
Page 35: Table VI: Convergence of the dissip

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