Weighted ENO Schemes for Hamilton--Jacobi Equations

Jiang, Guang-Shan; Peng, Danping

doi:10.1137/s106482759732455x

Cited by 996 publications

(798 citation statements)

References 17 publications

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“…We discretize the spatial operator |▽ϕ| in (7) and (8) using the fifth-order weighted essentially non-oscillatory (WENO) method [27,28], and we discretize pseudo-time in (8) using the third-order total variation-diminishing Runge-Kutta method (TVD-RK) from [23] and [24]. Due to the computational cost and the complexity of our tumor system, we currently discretize time in (7) using a forward Euler algorithm and a small step size.…”

Section: Narrow Band/local Level Set Methodsmentioning

confidence: 99%

“…First, we discretize the jump boundary condition by (27) The −sign(ϕ i,j ) term ensures that the jump condition has been applied in the proper direction from region Ω to region Ω c .…”

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

“…Because u r and v r are linearly independent, they form a basis for ℝ 2 , and we can write (32) where a r and b r are obtained by solving the linear system (33) Using this, we can express (∂p/∂n) r as a combination of grid (u r ) and off-grid (v r ) directional derivatives: (34) where (35) Using one-sided, first-order differences, we obtain the approximation (36) Similarly, u ℓ and v ℓ form a basis for ℝ 2 , and we can express (37) rewrite the normal derivative as (38) and approximate it (to first-order) as (39) By combining (36) and (39), we obtain a new discretization of [D∇p · n] = h: (40) When we combine this with the jump condition in (27), p ℓ and p r are completely determined. In practice, we implement this new discretization in the following way:…”

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

See 2 more Smart Citations

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasisteady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.

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Section: Narrow Band/local Level Set Methodsmentioning

confidence: 99%

“…First, we discretize the jump boundary condition by (27) The −sign(ϕ i,j ) term ensures that the jump condition has been applied in the proper direction from region Ω to region Ω c .…”

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

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A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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“…Osher and Fedkiw [11] introduced another method in this category called Level Set (LS). This method is not satisfying mass conservation even with the high resolution techniques [12], Therefore it is not so popular in Computational Fluid Dynamics. Different researchers try to combine methods to overcome the problem which was existing in each one individually.…”

Section: Introductionmentioning

confidence: 99%

Particle level set implementation on the finite volume method

Elmi

Kolahdouzan

2010

WIT Transactions on Engineering Sciences

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Prediction of free surface elevation is one of the most important phenomena in the field of free surface flow modelling. To date different procedures have been introduced to capture the free surface. In the current study, a EulerianLagrangian Method named Particle Level Set (PLS) is developed to predict the location of free surface. The first implementation of this scheme was on the Finite Difference Method. This paper deals with the implementation of Particle Level Set on the Finite Volume mesh. IN the PLS scheme, computational particles were deployed in conjunction with LS function to raise the accuracy of function values. Results obtained from the developed model were compared with experimental and numerical results of dam break cited in the literature which represent the capability of method to predict high gradient free surface changes accurately.

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“…The order of accuracy of the methods was increased through an essentially non-oscillatory (ENO) reconstruction in the upwind schemes of Osher, Sethian and Shu [17,18]. Weighted essentially non-oscillatory (WENO) reconstructions, which were first introduced for hyperbolic conservation laws [8,16], were then used by Jiang and Peng [7] to even further increase the accuracy of the numerical approximations using a compact reconstruction. Extensions of the first-order and ENO upwind methods to unstructured grids were done by Abgrall [1].…”

Section: Introductionmentioning

confidence: 99%

Semi-discrete Schemes for Hamilton-Jacobi Equations on Unstructured Grids

Levy

Nayak

2004

Numerical Mathematics and Advanced Applications

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Summary. We present a new semi-discrete central scheme for approximating solutions of Hamilton-Jacobi equations on unstructured meshes. This scheme extends the numerical Hamiltonians of Kurganov et al. to unstructured grids. Similarly to the previous works on structured grids, a semi-discrete formulation of central schemes is made possible due to estimates of the local speeds of propagation. The consistency of the method is obtained following Abgrall's calculations for the consistency of an upwind Lax-Friedrichs scheme on unstructured grids. We conclude with comments on high-order reconstructions.

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Weighted ENO Schemes for Hamilton--Jacobi Equations

Cited by 996 publications

References 17 publications

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Particle level set implementation on the finite volume method

Semi-discrete Schemes for Hamilton-Jacobi Equations on Unstructured Grids

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