Adaptive Moving Mesh Methods by Weizhang Huang

By Weizhang Huang

Moving mesh equipment are a good, mesh-adaptation-based method for the numerical resolution of mathematical types of actual phenomena. at the moment there exist 3 major recommendations for mesh edition, specifically, to take advantage of mesh subdivision, neighborhood excessive order approximation (sometimes mixed with mesh subdivision), and mesh circulate. The latter kind of adaptive mesh procedure has been much less good studied, either computationally and theoretically.

This booklet is ready adaptive mesh iteration and relocating mesh equipment for the numerical answer of time-dependent partial differential equations. It offers a common framework and concept for adaptive mesh iteration and offers a finished remedy of relocating mesh tools and their simple parts, besides their program for a couple of nontrivial actual difficulties. Many particular examples with computed figures illustrate many of the tools and the results of parameter offerings for these equipment. The partial differential equations thought of are customarily parabolic (diffusion-dominated, instead of convection-dominated).

The wide bibliography presents a useful consultant to the literature during this box. each one bankruptcy comprises invaluable workouts. Graduate scholars, researchers and practitioners operating during this quarter will reap the benefits of this book.

Weizhang Huang is a Professor within the division of arithmetic on the collage of Kansas.

Robert D. Russell is a Professor within the division of arithmetic at Simon Fraser University.

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Example text

X,t) tends to ξ ∗ (x) exponentially as t → ∞. When ρ = ρ(x,t) depends upon time, the theorem implies that b 1/2 (ξ (x,t) − ξ ∗ (x,t))2 dx = O(τ).

A mesh xn+1 at the new time level is first generated using the mesh and the physical solution (xn , un ) at the current time level, and the solution un+1 is then obtained at the new time level. Note that this mesh xn+1 adapts only to the current solution un and thus lags in time. This will not generally cause much trouble if the time step is reasonably small or the solution does not have abrupt changes in time. If the lag of the mesh in time causes a serious problem, several iterations of solving for the mesh and the physical PDE at each new time level can be used (cf.

B= .  ..  ..  .    ..  . 44)         .    ∗  ..  ∗ 16 1 Introduction (a) Computed solution. (b) Mesh trajectories. 9 1 Fig. 0. (b) The corresponding mesh trajectories. 9 1 Fig. 8 The time step size used in the adaptive moving mesh solution of Burgers’ equation with ε = 10−4 and 61 points is plotted as function of time. The relative and absolute tolerances for the time step control are taken as rtol = 10−6 and atol = 10−4 , respectively, for the Matlab ODE solver “ode15i” (using a backward differentiation formula of order 5).

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