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A locally second order symmetric method for discontinuous solution of Poisson’s equation on uniform Cartesian grids. (English) Zbl 1519.76345

Summary: A new method is proposed for numerically solving the Poisson equation for non-continuous scalar fields on a uniform Cartesian grid. The sharp discontinuity in both the magnitude and the gradient of the scalar field normal to the interface is represented by the numerical solution with second order accuracy at the interface. This is achieved by setting up a composite solution, which is a weighted average of two fictitious scalar fields that together produce the required discontinuity within each interfacial grid cell. A smooth treatment of the Poisson coefficient in a narrow band around the interface allows sharp interfacial jumps to be expressed with second order accuracy on regular grid points around the interface using a standard signed distance function. Moreover, the jump in the gradient tangent to the interface is not needed to enforce the jump in the gradient normal to the interface. The resulting linear system is symmetric and leads to second order accurate solutions on grid points adjacent to the interface. The accuracy of the new framework is compared with other methods.

MSC:

76T99 Multiphase and multicomponent flows
65N06 Finite difference methods for boundary value problems involving PDEs

Software:

CSparse
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Full Text: DOI

References:

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