Hyperbolic systems of equations posed on erroneous curved domains

The effect of an inaccurate geometry description on the solution accuracy of a hyperbolic problem is discussed. The inaccurate geometry can for example come from an imperfect CAD system, a faulty mesh generator, bad measurements or simply a misconception. We show that inaccurate geometry descriptions might lead to the wrong wave speeds, a misplacement of the boundary conditions, to the wrong boundary operator and a mismatch of boundary data. The errors caused by an inaccurate geometry description may affect the solution more than the accuracy of the specific discretization techniques used. In extreme cases, the order of accuracy goes to zero. Numerical experiments corroborate the theoretical results. 1. Erroneous computational domain Consider the following hyperbolic system of equations, in two space dimensions, Wt + ÂWx + B̂Wy = 0, (x, y) ∈Ω, t ∈ (0, T ], LW = g(x, y, t), (x, y) ∈ δΩ, t ∈ (0, T ], W = f (x, y), (x, y) ∈Ω, t = 0, (1) in which the solution is represented by the vector W = W (x, y, t). Â and B̂ are constant symmetric M×M matrices, Ω is the spatial domain with the boundary δΩ. The boundary operator L is defined on δΩ, f (x, y)∈RM and g(x, y, t)∈RM are the data in the problem. Preprint submitted to Journal of Computational Physics January 13, 2016 Equation (1) is transformed to curvilinear coordinates (ξ , η) by (Vξ , Vη , Vt)= [J](Vx, Vy, Vt) , where [J] is the Jacobian matrix of the transformation. The transformed problem is JWt +AWξ +BWη = 0, (ξ , η) ∈Φ, t ∈ (0, T ], LW = g(ξ , η , t), (ξ , η) ∈ δΦ, t ∈ (0, T ], W = f (ξ , η), (ξ , η) ∈Φ, t = 0, (2) where A = JξxÂ+ JξyB̂, B = JηxÂ+ JηyB̂ and J = xξ yη − xηyξ > 0 is the determinant of [J]. The energy method together with the metric identities and the use of the Green-Gauss theorem yields d dt ||W (ξ , η , t)||J =− ∮