Total variation diminishing Runge-Kutta schemes

Gottlieb, Sigal; Shu, Chi‐Wang

doi:10.1090/s0025-5718-98-00913-2

Cited by 2,088 publications

(1,227 citation statements)

References 16 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%

“…3. If (x i−2 , y j ) and (x i−1 , y j ) are both in Ω, then we obtain p̂i +1,j as a quadratic extrapolation of p from p i−2, j , P i−1,j , and u ℓ : (23) If (x i−1 , y j ) ∈ Ω but (x 1−2 , y j ) ∉ Ω, then we obtain p̂i +1,j by linear extrapolation from p i−1,j and p ℓ : (24) If (x i−1 , y j ) ∉ Ω, then we use the constant extrapolation p̂i +1,j = p ℓ . Note that in all cases, we require p ℓ .…”

Section: Ghost Cell Extrapolations For the Diffusionalmentioning

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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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%

Section: Ghost Cell Extrapolations For the Diffusionalmentioning

confidence: 99%

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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“…However, a possible consequence of using the higher-order schemes is the numerical instability. The Runge-Kutta methods with the total variation diminishing (TVD) property can prevent such a problem, see [34]. Considering the one-dimensional wave equation (24), the total variation of the variable ϕ can be calculated by,…”

Section: Time Integrationmentioning

confidence: 99%

“…It can be shown that the 1 st -order forward (Euler) time integration method is a basic TVD Runge-Kutta method, see [34]. The Euler method can be written as,…”

Section: • No Decreasing Values Of the Local Minimums And No Increasimentioning

confidence: 99%

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Level Set Method for Simulating the Interface Kinematics: Application of a Discontinuous Galerkin Method

Mousavi¹,

Kummer²,

Oberlack³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Finite Volume Methods: Foundation and Analysis

Barth

Ohlberger

2004

Encyclopedia of Computational Mechanics

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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

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Total variation diminishing Runge-Kutta schemes

Cited by 2,088 publications

References 16 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

Level Set Method for Simulating the Interface Kinematics: Application of a Discontinuous Galerkin Method

Finite Volume Methods: Foundation and Analysis

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