High order numerical methods for PDEs on complex domains and their applications in computational biology

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Efficient numerical methods have become more and more important for studying PDE models in computational biology. These time dependent PDE models can be parabolic or hyperbolic, and are defined on complex spatial domains. The PI proposes to perform research in the design, analysis of high order weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin (DG) methods on two-dimensional (2D) and three-dimensional (3D) unstructured meshes, for hyperbolic systems of conservation laws and stiff reaction-diffusion equations. And the PI proposes to apply these methods to high dimensional models for morphogen systems in developmental biology. Understanding morphogen gradient formation during embryo development is a fundamental problem in developmental biology. Mathematical models the PI will study are built on three-dimensional complex geometrical domains. Computational analysis of these models will reveal the formation mechanisms of BMP morphogen gradients, and the importance of incorporating 3D realistic shape of the embryo. In the spatial direction, the morphogen gradient systems usually develop sharp gradients on the embryo with a 3D complex shape. Hence the high order robust numerical methods on 2D and 3D unstructured meshes will be especially useful in the numerical simulations to deal with the 3D complex geometrical domains and the sharp gradients of the solutions. Four specific tasks are: 1) Develop high order WENO schemes and codes on 3D tetrahedral meshes; 2) Develop high order implicit integrating factor discontinuous Galerkin (IIF-DG) schemes and codes on 2D and 3D unstructured meshes; 3) Develop 3D reaction-diffusion models for BMP gradient formation during dorsal-ventral patterning of the zebrafish embryo. Starting from a simple model based on simplified bio-chemical gene network, the PI will create a more complete model motivated by new biological experiments; 4) Via computational analysis of the models on 3D complex geometrical domains, study the formation mechanisms of morphogen gradients, address important biological questions. Then, apply the model to other organisms such as Xenopus. The proposed numerical methods will be applied in the simulations. The proposed research will result in a suite of nice modeling and computational techniques, suitable for quantitative study of various morphogen systems, during the embryo development of different organisms. These techniques are expected to make positive contributions to computer simulations of complicated phenomena in morphogen gradient pattern formation. The proposed activity will also provide excellent training and education opportunities for both graduate and undergraduate students interested in research at the interface of mathematics, computation, and biology. The students will be well prepared to work in interdisciplinary research through developing intellectual and technical tools for the advancement of computational mathematics and applying these tools to important questions in biology.