High-Order, Finite-Volume Methods in Mapped Coordinates

4.4.2 Uniform Advection of Constant DataWe consider uniform advection of constant initial data on a periodic domainx ∈ [0, 1) 2 . The initial conditions areu 0 (x) = 1,(IC1)and the uniform velocity vector isv(x) = (1, 0.5). (86)The exact solution is u(x, t) = 1 at all times; we take a constant nominal CFLnumber of σ = v 1 ∆tN = 0.32 and integrate to a final time of t = 2. Whilethis problem appears to be trivial, it is an important demonstration of thefreestream preservation capability of the discretization.Results of grid convergence studies are presented in Tables 2 and 3. Freestreampreservation is demonstrated by the fact that the maximum pointwise error inthe computed solution in both cases is at most dominated by round-off error.Both with and without the limiter, the method is shown to be freestreampreserving.We note that the results differ with and without the limiter because the limiterscheme operates on data even when the solution is constant. The limiter neverreduces the order of the scheme in this case, but the method does a number offinite-precision operations that can introduce additional round-off error. Thisis particularly true on the deformed mesh (84); note that the non-constantJacobian for this mapping also contributes to round-off error. The magnitudeof round-off error increases with the number of cells because more time steps,and hence more floating-point operations, are required to reach the final time.If one were so inclined, a small modification to the limiter algorithm can bemade to prohibit it from acting on digits with dubious significance. Specifically,the first step in constructing face values and in constructing parabolic interpolantsis to compute a pair of differences between face and cell values. If themaximum relative magnitude of the two differences is above some small value(say 10 −15 ), the limiter algorithm proceeds normally; otherwise, the limitingprocess is skipped. In practice, this produces limited freestream-test resultsidentical to those for the base scheme (Table 2) with no apparent effect onlimiter performance for less trivial applications.27