A geometric approach to error estimates for conservation laws posed on a spacetime

Abstract

a b s t r a c tWe consider a hyperbolic conservation law posed on an (N + 1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L 1 error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.

Geometric error of finite volume schemes for conservation laws on evolving surfaces

“…As the directional derivative of d with respect to (1, 0, 0), (0, 1, 0) coincides with (ν Γ ) 1 , (ν Γ ) 2 , respectively, this proves (19). This immediately implies (ν Γ ) 3…”

“…In the previous error analysis for finite volume schemes approximating nonlinear conservation laws on manifolds the schemes were defined on curved elements lying on the curved surface and it was assumed that geometric quantities like lengths, areas and conormals are known exactly. While this is a reasonable assumption for schemes defined on general Riemannian manifolds or even more general structures [3,21] with no ambient space, most engineering applications involve equations on hypersurfaces of R 3 and one aims at computing the geometry with the least effort. This is in particular important for moving surfaces where the geometric quantities have to be computed in each time step.…”

“…We will treat this question in the "embedded" case where an explicit embedding of the surface under consideration into Euclidean space is known. It is an interesting question for future studies whether our analysis can be extended to errors arising from the discretisation of the geometry of the underlying space in an "invariant" description, like the very general one analysed in [3,21].…”

Global existence of smooth solutions to 2D Chaplygin gases on curved space

“…Here, one can refer to Dafermos and Rodnianski [17,18], Luk [19], Yang [20] and so on. On the other hand, the study of hyperbolic conservation laws on manifolds is another hot topic; one can refer to P. G. LeFloch [21,22].…”

Traces for Functions of Bounded Variation on Manifolds with Applications to Conservation Laws on Manifolds with Boundary

“…To our knowledge, conservation laws on manifolds were studied for the first time by [24] showing existence and uniqueness for a geometryindependent variant of (1.1). For the case of compact manifolds without boundary the theoretical and numerical study of conservation laws on manifolds has reached significant progress [8,2,7,3,19,11,20,13,14] during the last decade. The Dirichlet problem for a geometry-independent formulation of (1.1) was addressed by Panov using a kinetic formulation.…”