Convergence of Rothe's method for fully nonlinear parabolic equations

Blank, Ivan; Smith, Penelope

doi:10.1007/bf02930976

Cited by 4 publications

(2 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…as the approximation 5 of U (x, mτ, ξ), accounting for the choices of both τ and m (the same notation as used in [4]). We want to show that the L 1 (Ω n ) limit, U(x, t, ξ), of the sequence z m,t/m m∈N exists.…”

Section: Existencementioning

confidence: 99%

See 1 more Smart Citation

Computational framework to capture the spatiotemporal density of cells with a cumulative environmental coupling

Yereniuk

Olson

2020

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Stochastic agent-based models can account for millions of cells with spatiotemporal movement that can be a function of different factors. However, these simulations can be computationally expensive. In this work, we develop a novel computational framework to describe and simulate stochastic cellular processes that are coupled to the environment. Specifically, through upscaling, we derive a continuum governing equation that considers the cell density as a function of time, space, and a cumulative variable that is coupled to the environmental conditions. For this new governing equation, we consider the stability through an energy analysis, as well as proving uniqueness and wellposedness. To solve the governing equations in free-space, we propose a numerical method using fundamental solutions. As an application, we study a cell moving in an infinite domain that contains a toxic chemical, where a cumulative exposure above a critical value results in cell death. We illustrate the validity of this new modeling framework and associated numerical methods by comparing the density of live cells to results from the corresponding agent-based model.

show abstract

Section: Existencementioning

confidence: 99%

“…We solve the system in ( 14) using the method of Green's functions 4 and convoluting with the initial condition:…”

Section: Operator-splitting Semi-discrete Solutionmentioning

confidence: 99%