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We study homogenization of G-equation with a ow straining term (or the strain G-equation) in two dimensional periodic cellular ow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton-Jacobi equation. The main objective is to investigate how the ow induced straining (the nonconvex term) in uences front propagation as the ow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent ame speed sT (effective Hamiltonian) is well-defined for any periodic ow with small divergence and is enhanced by the cellular ow as $s_T \ge O(A/logA)$. In the second regime where A is slightly above the critical value, homogenization breaks down, and $s_T$ is not well defined along any direction. Solutions become a mixture of fast moving part and a stagnant part. When $A$ is sufficiently large, the whole ame front ceases to propagate forward due to the flow induced straining. In particular, along directions $p = (1; 0)$ and $(0;1)$, $s_T$ is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player dierential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX − XD − AX + B = 0, with M ≡ [D,−C;−B,A] ∈ R(n1+n2)×(n1+n2) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A−1u, A−⊤u, D−1v, and D−⊤v computable in O(n) complexity (with n = max{n1, n2}), for some vectors u and v, and B, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu [Numer. Math., 103 (2006), pp. 392–412] is adapted, with the appropriate applications of the Sherman–Morrison– Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.

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No abstract uploaded!