Abstract
This chapter extends the ideas in earlier chapters and identifies two concepts that are useful for checking the well-posedness of boundary value problems. These concepts play a fundamental role in establishing the stability of finite difference solutions in later chapters.
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Notes
- 1.
The alternative norm \(\Vert \mathscr {F}\Vert =\max _{(x,t)\in {\varOmega _{\tau }}}|f(x,t)|+\max _{(x,t)\in \varGamma _{\!\tau }}|g(x,t)|\) leads to a slightly larger upper bound.
- 2.
This might occur, for instance, when u is the pressure in a fluid. In such systems it is usually only the difference in pressure between two points and not the absolute pressure that can be measured.
- 3.
The PDE may be written in the form of a conservation law \(u_{t}+f(u)_{x}=0\) with a flux function \(f(u) = 2u-u_{x}\). The boundary conditions are then seen to be zero-flux conditions.
- 4.
An indication of the special nature of the KdV equation is that it has an infinite number of such conserved quantities.
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© 2015 Springer International Publishing Switzerland
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Griffiths, D.F., Dold, J.W., Silvester, D.J. (2015). Maximum Principles and Energy Methods. In: Essential Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-22569-2_7
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DOI: https://doi.org/10.1007/978-3-319-22569-2_7
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