Finite difference stencil

As stated above, the spatial derivative is approximated without regard to the time level. The distinction between explicit and implicit solutions depends on the time level at which the spatial derivatives are evaluated. Finite-difference stencils for explicit and implicit Euler methods are illustrated in Fig. 4.13. 

Fig. 4.13 Finite-difference stencils for the explicit and implicit Euler methods. The spatial index is j and the time index is n. For equally spaced radial mesh intervals of dr, rj = (j — 1 )dr, 1 < j < J. For equally spaced time intervals, tn = (n — 1 )dt, n > 1.

Fig. 6.11 Finite difference stencil for representing the stagnation-flow equations.

Figure 8. Finite difference grid with 5-point stencil.

Write finite-difference approximations to the governing equations. In deciding on the difference approximations, be sure to consider the boundary-condition information and the need for upwind differences. Sketch the difference scheme using a stencil. 

Figure 6.5 illustrates a stencil for the finite-difference discretization described by Eqs. 6.42 and 6.43. A spreadsheet that implements the finite-difference solution is described in detail in Appendix D (Section D.2). 

Fig. 6. 5 A stencil that illustrates the finite-difference discretization of the semi-infinite-domain axisymmetric stagnation flow problem.

Fig. 17.14 Finite-volume, staggered-grid, spatial-difference stencil for the transient compressible stagnation-flow equations. Grid points, which are at control-volume centers, are used to represent all dependent variables except axial velocity, which is represented at the control-volume faces. The grid indexes are shown on the left and the face indexes on the right. The right-facing protuberance on the stencils indicates where the time derivative is evaluated. For the pressure-eigenvalue equation there is no time derivative.

The use offictitious points as a means of locally modifying the differential stencils near laborious media interfaces in finite-difference simulations has been initially developed in [17, 18] and extended in [21, 28]. The specific method, which matches the problematic boundaries with physical derivative conditions, enhances the flexibility of higher order FDTD schemes and facilitates the discretization of difficult geometries. 

G. Sun and C. W. Trueman, Optimized finite-difference time-domain methods based on the (2,4) stencil, IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 832-842, Mar. 

Finite Volume and Finite Difference Methods for Modeling and Simulation, Fig. 1 A generic finite volume discretization stencil in two dimensions... 

The normal vector n, is defined by the vector gradient of Cg, which can be derived from different finite difference approximations which directly influence the accuracy of algorithms. These include Green-Gauss, volume-average, least-squares, minimization principle, and Young s gradients. It is noted that a wide, symmetric stencil for n,y is necessary for a reasonable estimation of the interface orientation. 

FIGURE 7 A finite-difference grid decomposed among the nodes of a concurrent processor. Each processor is responsible for a 4x4 subgrid. Laplace s equation is to be solved for 0 using a simple relaxation technique, Eq. (4). The five-point update stencil is also shown. 

This quantity is computed on a smoothed /-field /, which is obtained by convo-lutmg/with a B-spline of degree 2. The normal computed with an 18-points stencil at the center of the cell face. These face-centered gradients of / are then used to calculate the capillary stress tensor at the cell faces. Central differences yield the actual surface tension force in the cell centers. This surface tension force is then evenly distributed to the cell faces, where the velocities are stored. A detailed description of the above methods and the finite-volume method for fluid dynamics used in FS3D can be found in [25]. 

---