Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Monotone Mixed Finite Difference Scheme for Monge-Amp\`ere Equation   In this paper, we propose a monotone mixed finite difference scheme for
solving the two-dimensional Monge-Amp\`ere equation. In order to accomplish
this, we convert the Monge-Amp\`ere equation to an equivalent
Hamilton-Jacobi-Bellman (HJB) equation. Based on the HJB formulation, we apply
the standard 7-point stencil discretization, which is second order accurate, to
the grid points wherever monotonicity holds, and apply semi-Lagrangian wide
stencil discretization elsewhere to ensure monotonicity on the entire
computational domain. By dividing the admissible control set into six regions
and optimizing the sub-problem in each region, the computational cost of the
optimization problem at each grid point is reduced from $O(M^2)$ to $O(1)$ when
the standard 7-point stencil discretization is applied and to $O(M)$ otherwise,
where the discretized control set is $M \times M$. We prove that our numerical
scheme satisfies consistency, stability, monotonicity and strong comparison
principle, and hence is convergent to the viscosity solution of the
Monge-Amp\`ere equation. In the numerical results, second order convergence
rate is achieved when the standard 7-point stencil discretization is applied
monotonically on the entire computation domain, and up to order one convergence
is achieved otherwise. The proposed mixed scheme yields a smaller
discretization error and a faster convergence rate compared to the pure
semi-Lagrangian wide stencil scheme.

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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga