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A Taylor series analysis of the upwind scheme discussed above will show that it is first-order accurate in space and time. Modified wavenumber analysis shows that the first-order upwind scheme introduces severe numerical diffusion /dissipation in the solution where large gradients exist due to necessity of high wavenumbers to represent sharp ...
Second, a mathematical structure may have more than one definition (for example, topological space has at least seven definitions; ordered field has at least two definitions). In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the ...
A geometry: it is equipped with a metric and is flat. A topology: there is a notion of open sets. There are interfaces among these: Its order and, independently, its metric structure induce its topology. Its order and algebraic structure make it into an ordered field.
Let (,) and (,) be ordered pairs. Then the characteristic (or defining) property of the ordered pair is: (,) = (,) = =.. The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B, and written A × B.
However, for large Peclet numbers (generally > 2) this approximation gave inaccurate results. It was recognized independently by several investigators [1] [2] that the less expensive but only first order accurate upwind scheme can be employed but that this scheme produces results with false diffusion for multidimensional cases. Many new schemes ...
Shows the analytical solution along with a simulation based upon a first order upwind spatial discretization scheme. We will consider the fundamentals of the MUSCL scheme by considering the following simple first-order, scalar, 1D system, which is assumed to have a wave propagating in the positive direction,
Following the classical finite volume method framework, we seek to track a finite set of discrete unknowns, = / + / (,) where the / = + (/) and = form a discrete set of points for the hyperbolic problem: + (()) =, where the indices and indicate the derivatives in time and space, respectively.
There is more than one possible semantics for higher-order logic, as discussed in the article on second-order logic. When using full higher-order semantics, a structure need only have a universe for objects of type 0, and the T-schema is extended so that a quantifier over a higher-order type is satisfied by the model if and only if it is ...