We introduce a new nonlocal calculus framework which parallels (and includes as a limiting case) the differential setting. The integral operators introduced have convolution structures and converge as the horizon of interaction shrinks to zero to the classical gradient, divergence, curl, and Laplacian...
Long-time regularized shallow water simulations show that the VFL method exhibits no secular drift in the (i) energy error through the application of symplectic integrators; and (ii) the potential vorticity error through the construction of discrete curl, divergence and gradient operators which ...
In this paper, we develop high order interpolants, from any basis that constitutes a partition of unity, which satisfy these integral relations exactly. The resulting gradient, curl and divergence conforming spaces have the...doi:10.1007/978-3-319-01601-6_23René Hiemstra...
The resulting gradient, curl and divergence conforming spaces have the property that the conservation laws become completely independent of the basis functions. Hence, they are exactly satisfied at the coarsest level of discretization and on arbitrarily curved meshes. As an illustration we apply our ...
These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as...
Specifically, in the EPT methods of Section 4.4 and Section 4.5, the generalized Helmholtz equation is rewritten in terms of the gradient of B ^ 1 + and B ^ z , while, in the methods of Section 4.6, the generalized Helmholtz equation is written as a convection–reaction equation. 3.2.1...