High-Order, Finite-Volume Methods in Mapped Coordinates

10 0 Uniform Advection of Compact Data (IC2)10 −110 −210 −3Max Norm of Error10 −410 −510 −6Cartesian: Base10 −7Deformed: BaseCartesian: Limiter10 −8 Deformed: LimiterFirst−OrderFourth−Order10 −910 1 10 2 10 3Number of CellsL ∞ Error Norm for Base SchemeN cells 16 32 64 128 256 512CartesianDeformederror 5.12e-01 2.81e-01 4.14e-02 2.81e-03 1.78e-04 1.12e-05rate - 0.87 2.76 3.88 3.98 4.00error 3.26e-01 3.11e-01 1.04e-01 7.38e-03 4.70e-04 2.96e-05rate - 0.07 1.59 3.81 3.97 3.99L ∞ Error Norm for Base Scheme with limiterN cells 16 32 64 128 256 512CartesianDeformederror 4.52e-01 2.38e-01 3.33e-02 2.82e-03 1.78e-04 1.12e-05rate - 0.93 2.84 3.56 3.98 4.00error 4.19e-01 3.77e-01 6.66e-02 7.44e-03 4.70e-04 2.96e-05rate - 0.15 2.50 3.16 3.98 3.99Fig. 12. Convergence of maximum pointwise error with grid refinement for the baseand limiter schemes for uniform advection (86) of compactly-supported data (IC3)on both meshes.in Figure 13. The basic shape of the cylinder is preserved, albeit with somespreading of the initially sharp edges of the cylinder. The slot has some fill-in,but the shape and size of the slot is fairly well preserved. This is a very goodapproximation for such a coarse resolution; note that there are only five cellsacross the slot.The initial and final solution for simulation on the mapped grid (84) are plottedin Figure 14. Again, the basic shape of the cylinder is preserved, althoughthere is more smearing than for the Cartesian case. In addition, note thatthere are slightly larger overshoots (still only 1%) than for the Cartesian case.The symmetry of the cylinder is still relatively good, despite the distortions32