HPC Tier Module
Tumor Invasion (Fisher-KPP Reaction-Diffusion)
Propagate a Fisher-KPP invasion front and verify its speed against the exact c = 2*sqrt(rD) - in the browser.
This tool runs inside GDBS HPC. What it computes and the authorities it applies are below.
The problem
The mathematics of invasion fronts is taught with simulations whose front speed is rarely checked against the exact KPP law, leaving students and modelers unable to tell a correct solve from a plausible-looking one.
The local-first solution
This plugin marches the cited Fisher-KPP equation by explicit or IMEX finite differences in exact f64, measures the front position over time, and fits its speed against the closed-form minimal wave speed - all deterministic and in your browser with no upload.
What it does
Method-of-lines solve with a centred 3-point Laplacian and a logistic reaction source
Explicit forward-Euler or IMEX (implicit diffusion, explicit reaction) integrators
Diffusion Fourier number and reaction step number computed and stability-flagged
Measures the front position at a level set and fits its propagation speed
Checks the speed against the closed-form c_min = 2*sqrt(rD)
Verifies saturation at the carrying capacity behind the front
Honest scope
Exact: the 3-point Laplacian, the forward-Euler and backward-Euler/Thomas steps, the logistic source, the Fourier-number stability test, and the closed-form minimal wave speed it is checked against. The model uses a single dimensionless density with constant isotropic diffusion and growth, logistic saturation, and 1D geometry; whether D and r match a particular tumor is a separate calibration question. It does not model anisotropic or spatially varying diffusion, mechanical confinement, angiogenesis, treatment terms, multi-species structure, or 2D/3D anatomy. Research and education only - not a clinical, diagnostic, prognostic, or treatment-planning device, and not for medical decisions.
Authorities cited
- Fisher, R. A. (1937). The Wave of Advance of Advantageous Genes. Annals of Eugenics 7(4), 355-369. - the reaction-diffusion equation u_t = D u_xx + r u(1-u) and its traveling-wave fronts. DOI 10.1111/j.1469-1809.1937.tb02153.x.
- Kolmogorov, A. N., Petrovsky, I. G., & Piskunov, N. S. (1937). A study of the diffusion equation with increase in the amount of substance. Moscow Univ. Bull. Math. 1, 1-25. - the KPP analysis selecting the MINIMAL front speed c = 2 sqrt(rD) for steep / compactly-supported initial data (the analytic benchmark used here).
- Murray, J. D. (2003). Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd ed., Ch. 13 (Fisher-KPP wave fronts) and Ch. 11 (gliomas). Springer. - the c = 2 sqrt(rD) minimal-speed selection and the tumor-growth application. DOI 10.1007/b98869.
- Swanson, K. R., Alvord, E. C., & Murray, J. D. (2000). A quantitative model for differential motility of gliomas in grey and white matter. Cell Proliferation 33(5), 317-329. - the Fisher-KPP (reaction-diffusion) continuum model of glioblastoma invasion with diffusion D and net proliferation rate rho. DOI 10.1046/j.1365-2184.2000.00177.x.
- LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM. sec 1.3 - the centred 2nd-order 3-point Laplacian; ghost-node Neumann BCs (sec 2.12). DOI 10.1137/1.9780898717839.
- Strikwerda, J. C. (1989/2004). Finite Difference Schemes and Partial Differential Equations, 2nd ed. SIAM. - von Neumann stability of the explicit (FTCS) diffusion step; the 1D limit D dt/dx^2 <= 1/2. DOI 10.1137/1.9780898717938.
- Ascher, U. M., Ruuth, S. J., & Wetton, B. T. R. (1995). Implicit-Explicit Methods for Time-Dependent PDEs. SIAM J. Numer. Anal. 32(3), 797-823. - the IMEX (implicit diffusion / explicit reaction) splitting used here. DOI 10.1137/0732037.
- Press, W. H., et al. (2007). Numerical Recipes, 3rd ed. CUP. sec 2.4 - the Thomas algorithm for tridiagonal systems.
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Run the invasion solve
Compute the front and its wave-speed error in your browser, with nothing uploaded to anyone's cloud. Save the run to Sandbox, attach it to a Worklog case, or share parameters through a Gate portal.