High-Order, Finite-Volume Methods in Mapped Coordinates

Magnitude of Amplification FactorRelative Phase Speed10.99σ=1.00σ=0.75σ=0.50σ=0.2510.80.6|g|0.98α/a0.40.970.20.960 0.5 1θ/π00 0.5 1θ/πFig. 7. Variation of the magnitude of the amplification factor |g| and the relativephase speed α/a with phase angle θ for several values of σ. Note that as θ → ±π,the damping vanishes and the modes do not propagate. Furthermore, the phaseerror is effectively independent of σ.where the discrete phase angles are θ id = 2πi d /n, i d = 0, ±1, ±2, . . . , ±n/2.Because of the central spatial discretization, the eigenvalues are all pure imaginary,that is, the spatial discretization contributes no numerical dissipation.The magnitude of the amplification factor as a function of continuous phaseangles is( )|g| = √ 1 − y61 − y2. (80)72 8For D = 1, this is plotted in Figure 7. Similarly, the relative phase speed ofthe one-dimensional scheme,α(θ)a= − 1 Im g(θ)σθ Re g(θ) , (81)where σ = a∆t/h, is also plotted in Figure 7. We see that the Runge Kuttascheme adds a small amount of dissipation and that, as θ → ±π, this dissipationvanishes. At the same time, we see that these high-wavenumber modes(grid modes) do not propagate. For variable-coefficient and nonlinear problems,these undamped grid modes can pollute the solutions, if not cause instability.4.3 LimitingOne approach to stabilize a high-order, no- or low-dissipation scheme forvariable-coefficient or nonlinear hyperbolic problems is to add artificial dis-23