In this article, we study initial and initial-boundary value problems for a non-strictly hyperbolic system whose characteristic speed is not smooth and takes values in {−1, 0, 1}. We construct an explicit formula for the weak solution.We also study the interaction of waves and the large time asymptotic behavior of a solution for the case when the initial data is periodic with zero mean over the period and also for the case when the initial data has compact support.
We analyze the convergence of discretization schemes
for the adjoint equation arising in the adjoint-based derivative computation
for optimal control problems governed by entropy solutions
of conservation laws. The difficulties arise from the fact that the correct
adjoint state is the reversible solution of a transport equation
with discontinuous coefficient and discontinuous end data. We derive
the discrete adjoint scheme for monotone difference schemes in
conservation form. It is known that convergence of the discrete adjoint
can only be expected if the numerical scheme has viscosity of
order O(h) with appropriate 0 < α < 1, which leads to quite viscous
shock profiles. We show that by a slight modification of the
end data of the discrete adjoint scheme, convergence to the correct
reversible solution can be obtained also for numerical schemes with
viscosity of order O(h) and with sharp shock resolution. The theoretical
findings are confirmed by numerical results.
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Let lj := -d2 /dx2 + k2qj(x), k = const > 0, j = 1,2, 0 < ess inf qj(x) ≤ ess sup qj(x) < ∞. Suppose that (∗) [integral of, between limits 1 and 0] p(x)u1(x, k)u2(x, k) dx = 0 for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0, 1], which changes sign finitely many times, and uj solves the problem ljuj = 0, 0 ≤ x ≤ 1, u'j(0, k) = 0, uj(0, k) = 1. It is proved that (∗) implies p = 0. This result is applied to an inverse problem for a heat equation.
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