Discrete Traveling Waves Which Approximate Shocks
Discrete Traveling Waves Which Approximate Shocks
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We set u = and solve for u-, X o 1 u^(e) = + i^4^ • (3. 3) We choose the root so that u-, (e) < 0. The Courant-Friedrichs-Lewy condition for this equation is _< -p. As varies from to -^, u-, (e) varies from to - /^+1 and is monotone in 0o This substantiates the empirical observation that the profile of a traveling wave becomes steeper as 6 is increased to the limit imposed by the Courant-Friedrichs-Lewy condition. We do not establish this result in general. We establish a weaker result. We show that the traveling waves we have constructed converge to a monotone profile as v\. Runs through rational values to an irrational limit. The limiting shape is defined on the entire real line and satisfies ^x+n = ^(^x+k^^x+k-i---^x-k^ ^^-"^^ and has limits u, = u, and u = u^. + 00 t' - go H, 25 When T^ is rational the minimal domain over which O-^) makes sense is the linear span over the integers of 1 and t]. The distance between adjacent points in the lattice is a function of r|, which we call A(t]).
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You can find similar books in the "Read Also" column, or choose other free books by Gray Jennings to read online
To quickly assess the difficulty of the text, read a short excerpt:
We set u = and solve for u-, X o 1 u^(e) = + i^4^ • (3. 3) We choose the root so that u-, (e) < 0. The Courant-Friedrichs-Lewy condition for this equation is _< -p. As varies from to -^, u-, (e) varies from to - /^+1 and is monotone in 0o This substantiates the empirical observation that the profile of a traveling wave becomes steeper as 6 is increased to the limit imposed by the Courant-Friedrichs-Lewy condition. We do not establish this result in general. We establish a weaker result. We show that the traveling waves we have constructed converge to a monotone profile as v\. Runs through rational values to an irrational limit. The limiting shape is defined on the entire real line and satisfies ^x+n = ^(^x+k^^x+k-i---^x-k^ ^^-"^^ and has limits u, = u, and u = u^. + 00 t' - go H, 25 When T^ is rational the minimal domain over which O-^) makes sense is the linear span over the integers of 1 and t]. The distance between adjacent points in the lattice is a function of r|, which we call A(t]).
What to read after Discrete Traveling Waves Which Approximate Shocks?
You can find similar books in the "Read Also" column, or choose other free books by Gray Jennings to read online
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