High-Order Discontinuous Galerkin Simulations of a Reaction–Diffusion–Haptotaxis Model for Tumor Invasion

Cancer invasion couples tumor-cell proliferation, haptotactic migration, extracellular matrix (ECM) degradation, and matrix-degrading enzyme (MDE) dynamics. We develop a high-order modal discontinuous Galerkin (DG) method for a reaction–diffusion–haptotaxis model of these processes. The scheme combines a hierarchical modal Legendre basis, a Bassi–Rebay 2 (BR2) discretization of the diffusion terms, a central haptotactic flux based on a reconstructed DG gradient of the ECM density, and third-order strong-stability-preserving Runge–Kutta time integration. Manufactured-solution tests show the expected spatial orders for polynomial degrees p=1,2,3 (approaching fourth order for p=3) and third-order temporal accuracy, and the method reproduces published non-remodeling benchmark results. One-dimensional studies, supported by mesh and time-step sensitivity tests, show that a larger proliferation rate accelerates tumor expansion, ECM depletion, and MDE accumulation, and that haptotaxis advances the invasion front and depletes the ECM earlier. The scheme is not positivity preserving and uses no limiter. The undershoot of the tumor-cell density is below 4×10−3 in the proliferation study but reaches 0.22 at the strongest haptotaxis considered (χ=0.02), so the haptotaxis study is restricted to weak-to-moderate values. A two-dimensional simulation illustrates the extension of the framework.