35L04 Initial-boundary value problems for first-order hyperbolic equations
Refine
Document Type
- Doctoral Thesis (1)
- Preprint (1)
Language
- English (2)
Has Fulltext
- yes (2)
Keywords
- asymptotic-preserving (1)
- glioblastoma (1)
- haptotaxis (1)
- kinetic equations (1)
- multi-scale model (1)
- numerics (1)
Faculty / Organisational entity
We study a multi-scale model for growth of malignant gliomas in the human brain.
Interactions of individual glioma cells with their environment determine the gross tumor shape.
We connect models on different time and length scales to derive a practical description of tumor growth that takes these microscopic interactions into account.
From a simple subcellular model for haptotactic interactions of glioma cells with the white matter we derive a microscopic particle system, which leads to a meso-scale model for the distribution of particles, and finally to a macroscopic description of the cell density.
The main body of this work is dedicated to the development and study of numerical methods adequate for the meso-scale transport model and its transition to the macroscopic limit.
A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.