By P Wesseling

ISBN-10: 0471930830

ISBN-13: 9780471930839

Multigrid tools have constructed quickly and are used as a strong device for the effective resolution of elliptic and hyperbolic equations. this article offers an advent to multigrid tools for partial differential equations, with functions to sensible move difficulties.

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**Additional info for An Introduction to MultiGrid Methods**

**Sample text**

Cn ∈ R and deﬁne a new function u(x, t) by n ck uk (x, t). 48) with initial data n ck f k (x). 17 Consider the function S(x, t) given by S(x, t) = √ 1 − x2 e 4t 4πt for x ∈ R, t > 0. This function is well known in probability theory. It corresponds to the density function for the normal distribution with variance 2t. As we shall see below, this function also appears naturally in the analysis of the Cauchy problem for the heat equation. In the context of diﬀerential equations the function S is therefore frequently referred to as the Gaussian kernel function or the fundamental solution of the heat equation.

C) Explain why t u(x, t) = φ(x − ct) + 0 v[x − c(t − τ ), τ ] dτ. 33) for u(x, t). 1. 359∆t. 18). Show that if tm = m∆t, then u(tm+1 ) − u(tm ) = u(tm ) + τm , ∆t where the truncation error τm satisﬁes |τm | ≤ ∆t T e 2 for 0 ≤ (m + 1)∆t ≤ T. (b) Assume that {vm } is the corresponding forward Euler solution given by v0 = 1, vm+1 = (1 + ∆t)vm , and let wm = um − vm be the error at time tm = m∆t. Explain why {wm } satisﬁes the diﬀerence equation wm+1 = (1 + ∆t)wm + ∆t τm , w0 = 0. (c) Use induction on m to prove that ∆t T tm e (e − 1) for 0 ≤ tm ≤ T.

Hence, the constants c1 and c2 are uniquely determined from the boundary conditions. 1) has a unique solution. 6), we are allowed to write the solution u in the following form: 1 u(x) = x 0 x (1 − y)f (y) dy − 0 (x − y)f (y) dy. 1) with f (x) = 1. 7) we easily obtain 1 u(x) = x 0 x (1 − y) dy − 0 (x − y) dy = 1 x(1 − x). 2 42 2. 1) with f (x) = x. 7) we get 1 u(x) = x 0 x (1 − y)y dy − 0 (x − y)y dy = 1 x(1 − x2 ). 6 Further examples of how to compute the exact solution formulas for twopoint boundary value problems are given in the exercises.

### An Introduction to MultiGrid Methods by P Wesseling

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Categories: Differential Equations