Download Topics in Numerical Partial Differential Equations and by Susanne C. Brenner PDF
By Susanne C. Brenner
Numerical partial differential equations (PDEs) are an enormous a part of numerical simulation, the 3rd part of the fashionable method for technology and engineering, in addition to the normal idea and scan. This quantity comprises papers that originated with the collaborative learn of the groups that participated within the IMA Workshop for girls in utilized arithmetic: Numerical Partial Differential Equations and clinical Computing in August 2014.
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The largest stable value of CFL for the different methods is given in Table 8. In the linear case, with a given reasonably small ε, the differences between the allowable time-step for the different WENO methods are not significant. However, the choice of time-stepping method makes a difference. The CFL for the ten-stage fourth-order RK10,4 is significantly larger than that of three-stage third-order RK3,3. Even when we take into account that RK10,4 requires ten stages and RK3,3 only three, we can conclude that RK10,4 has an advantage, because it allows for higher effective CFL numbers when taking into account the number of stages.
7) One can view this model as the model on RN by a zero extension of u(x, t) from Ω to RN \Ω. The model (1) with periodic boundary conditions is given by u t = d [τ u + (1 − τ )K u] + u(m(x) − u) for x ∈ R N , u(x, t) = u(x + p, t) for x ∈ R N , (8) where p = ( p1 , p2 , . . , p N ) is a constant vector and the condition u(x) = u(x + p) for x ∈ R N is the so-called p−periodic function. One can view this as a periodic extension from a finite domain Ω = (0, p1 ) × (0, p2 ) × · · · × (0, p N ) to R N .
To study this behavior on a smoother function, with only one discontinuity, we consider the case where the initial condition u0 is not a step-function: 42 B. Dong et al. 8 cfl Fig. 7 The coefficient Cm (CFL) of the undershoot of WENO-M and WENO-Z after one timestep. (a) WENO-M, f (u) = u. (b) WENO-Z, f (u) = u. (c) WENO-M, f (u) = u2 /2. (d) WENO-Z, f (u) = u2 /2. (e) WENO-M, f (u) = u3 /3. (f) WENO-Z, f (u) = u3 /3. (g) WENO-M, f (u) = u4 /4. (h) WENO-Z, f (u) = u4 /4. The Effect of the Sensitivity Parameter … 43 Table 5 The undershoot error of the WENO method after one time-step of the time-stepping method with the nonstep function initial condition (7).