Solving Poisson-type equations with Robin boundary conditions on piecewise smooth interfaces

Bochkov, Daniil; Gibou, Frédéric

doi:10.1016/j.jcp.2018.10.020

Cited by 39 publications

(26 citation statements)

References 81 publications

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“…, 9) calculated according to the above-mentioned procedure lead to a non-symmetric global system of semidiscrete equations. Non-symmetric global matrices are also reported for other numerical techniques with Cartesian meshes on irregular domains, e.g., see [6].…”

Section: Remarkmentioning

confidence: 98%

“…. , 6,8,9) and l i (i = 1, 2, 3) are to be determined from the minimization of the local truncation error, the expression in the square brackets in the right-hand side of Eq. (52) represents the Neumann boundary conditions at three boundary points with the coordinates…”

Section: Neumann Boundary Conditions (With No Inclusion Of Boundary Degrees Of Freedom)mentioning

confidence: 99%

“…. , 6, 8, 9) andl i (i = 1, 2, 3) of the stencil equation are: 6,8,9) andl i = l i /m 5 (i = 1, 2, 3), i.e., there are only 18 unknown rescaled coefficients. The case of nonzero loadf A,B = 0 can be similarly treated because the termf A,B is a linear function of the coefficients m i (see Eq.…”

Section: Neumann Boundary Conditions (With No Inclusion Of Boundary Degrees Of Freedom)mentioning

confidence: 99%

“…. , 6,8,9) in terms of m 1 , m 5 and l i (i = 1, 2, 3) and are given in the attached file 'm-k-coeff.pdf' where the coefficients m 5 can be taken as m 5 = 1 after rescaling; see Remark 9. The remaining unknown coefficients m 1 and l i (i = 1, 2, 3) can be found from the minimization of the leading terms of the local truncation error at the constraint b 12 = 0.…”

Section: Remark 10mentioning

confidence: 99%

See 3 more Smart Citations

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

Idesman

2020

Arch Appl Mech

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A new numerical approach for the time-dependent wave and heat equations as well as for the timeindependent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

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Section: Remarkmentioning

confidence: 98%

Section: Neumann Boundary Conditions (With No Inclusion Of Boundary Degrees Of Freedom)mentioning

confidence: 99%

Section: Neumann Boundary Conditions (With No Inclusion Of Boundary Degrees Of Freedom)mentioning

confidence: 99%

Section: Remark 10mentioning

confidence: 99%

See 2 more Smart Citations

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

Idesman

2020

Arch Appl Mech

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show abstract

“…Specifically, the truncation error 1 is O h 2 for grid points away from the immersed interface and O (1) for cells crossed by the interface. Following the results of [35,26,57,12,50,55,22,24,10], we expect the schemes to produce second-order accurate numerical solutions with first-order accurate gradients.…”

Section: Numerical Discretizationmentioning

confidence: 91%

Solving elliptic interface problems with jump conditions on Cartesian grids

Bochkov

Gibou

2020

Journal of Computational Physics

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We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces second-order accurate solutions and first-order accurate gradients in the L ∞norm on Cartesian grids. The condition number is bounded, regardless of the ratio of the diffusion constant and scales like that of the standard 5-point stencil approximation on a rectangular grid with no interface. Numerical examples are given in two and three spatial dimensions.

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A Fast Sine Transform Accelerated High-Order Finite Difference Method for Parabolic Problems over Irregular Domains

Ren

Long

et al. 2023

J Sci Comput

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Solving Poisson-type equations with Robin boundary conditions on piecewise smooth interfaces

Cited by 39 publications

References 81 publications

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

Solving elliptic interface problems with jump conditions on Cartesian grids

A Fast Sine Transform Accelerated High-Order Finite Difference Method for Parabolic Problems over Irregular Domains

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