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Finite difference methods for approximating Heaviside functions. (English) Zbl 1171.65014
The paper presents two algorithms to evaluate the integral $$\mathcal{I}:=\int_\Omega f(\vec{x})d\vec{x}$$, where $$\vec{x}\in \mathbb{R}^n$$, $$\Omega=\{\vec{x}:\,u(\vec{x})>0\}$$ and $$\partial\Omega=\{\vec{x}:\,u(\vec{x})=0\}$$ is a compact manifold of codimension one. The method to be employed is motivated by the expression $$\mathcal{I}=\int_{\mathbb{R}^n} H(u(\vec{x}))f(\vec{x})d\vec{x}$$, where $$H$$ is the Heaviside function. The approach consists of approximating $$H$$ by finite differencing its first few primitives, a technique already used by the author to approximate delta functions.
A brief presentation of the algorithms is given below. Let
$I(z)=\int_0^z H(\zeta)d\zeta\quad\text{and}\quad J(z)=\int_0^z I(\zeta)d\zeta.$
The following relationships are derived:
$I(u)=\langle\nabla J(u),\nabla u \rangle /|\nabla u|^2,\quad H(u)=\langle\nabla I(u),\nabla u \rangle /|\nabla u|^2,$
where $$\langle \cdot,\cdot \rangle$$ stands for the inner product. By discretizing $$H(u)$$ the one-step algorithm $$FDMH_1$$ is obtained which converges at a rate of $$\mathcal{O}(h^2)$$ when $$u$$ is smooth enough. By discretizing both relationships the two-step algorithm $$FDMH_2$$ is derived which can converge at a rate of $$\mathcal{O}(h^3)$$. These results are validated by means of some numerical examples.

##### MSC:
 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures 41A25 Rate of convergence, degree of approximation 41A63 Multidimensional problems
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