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Mathematics of Computation

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Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations

Author: William J. Layton
Journal: Math. Comp. 36 (1981), 87-92
MSC: Primary 65N30; Secondary 65M15
MathSciNet review: 595043
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Abstract: We consider the approximate solution of the initial value problem

$\displaystyle \frac{{\partial u}}{{\partial t}} = \frac{{\partial u}}{{\partial x}},\quad u(x,0) = v(x),$

by a dissipative Galerkin method. When v is taken to have a jump discontinuity at zero, that discontinuity will propagate along $ x + t = 0$, in the true solution u. Estimates in $ {L_2}$ and $ {L_\infty }$ of the pollution effects of the discontinuity are found. These estimates show those effects to decay exponentially in $ {h^{ - 1}}$ in regions a fixed distance d from the discontinuity and exponentially in d for fixed h.

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Article copyright: © Copyright 1981 American Mathematical Society

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