A probabilistic wide-stencil scheme for elliptic equations preserves positivity, remains unconditionally stable, and achieves O(h) or O(h^{1/2}) convergence depending on boundary conditions.
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A Non-compact Positivity-Preserving Numerical Scheme for Elliptic Differential Equations Based on Mathematical Expectation
A probabilistic wide-stencil scheme for elliptic equations preserves positivity, remains unconditionally stable, and achieves O(h) or O(h^{1/2}) convergence depending on boundary conditions.