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  2. First-order upwind scheme - Wikipedia

    en.wikipedia.org/wiki/Upwind_scheme

    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 ...

  3. MacCormack method - Wikipedia

    en.wikipedia.org/wiki/MacCormack_method

    The order of differencing can be reversed for the time step (i.e., forward/backward followed by backward/forward). For nonlinear equations, this procedure provides the best results. For linear equations, the MacCormack scheme is equivalent to the Lax–Wendroff method .

  4. MUSCL scheme - Wikipedia

    en.wikipedia.org/wiki/MUSCL_scheme

    In this paper he constructed the first high-order, total variation diminishing (TVD) scheme where he obtained second order spatial accuracy. The idea is to replace the piecewise constant approximation of Godunov's scheme by reconstructed states, derived from cell-averaged states obtained from the previous time-step. For each cell, slope limited ...

  5. QUICK scheme - Wikipedia

    en.wikipedia.org/wiki/Quick_scheme

    In order to find the cell face value a quadratic function passing through two bracketing or surrounding nodes and one node on the upstream side must be used. In central differencing scheme and second order upwind scheme the first order derivative is included and the second order derivative is ignored.

  6. Hybrid difference scheme - Wikipedia

    en.wikipedia.org/wiki/Hybrid_difference_scheme

    For large Peclet numbers (|Pe| > 2) it uses the Upwind difference scheme, which first order accurate but takes into account the convection of the fluid. As it can be seen in figure 4 that for Pe = 0, it is a linear distribution and for high Pe it takes the upstream value depending on the flow direction.

  7. Godunov's scheme - Wikipedia

    en.wikipedia.org/wiki/Godunov's_scheme

    Exact time integration of the above formula from time = to time = + yields the exact update formula: + = + (((, + /)) ((, /))). Godunov's method replaces the time integral of each ∫ t n t n + 1 f ( q ( t , x i − 1 / 2 ) ) d t {\displaystyle \int _{t^{n}}^{t^{n+1}}f(q(t,x_{i-1/2}))\,dt} with a forward Euler method which yields a fully ...

  8. Talk:Upwind scheme - Wikipedia

    en.wikipedia.org/wiki/Talk:Upwind_scheme

    Though, dissipation can be thought of as artificial (or numerical) diffusion. Running the Forward-Euler, first order Upwind scheme on a simple advection (first order wave) equation with a coefficient of unity (A=1) with a binary initial condition (either 0 or 1) and periodic boundary conditions will eventually lead to no gradients what-so-ever.

  9. Order of accuracy - Wikipedia

    en.wikipedia.org/wiki/Order_of_accuracy

    In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider u {\displaystyle u} , the exact solution to a differential equation in an appropriate normed space ( V , | | | | ) {\displaystyle (V,||\ ||)} .