Numerical Methods Of Solutions For Two Kinds Of Nonlocal Partial Differential Equations In Biological Evolution Model

This thesis investigates numerical methods for two kinds of nonlocal partial differential equations in biological evolution model. These integro–di?erential equations describe the evolution of a population structured with respect to a continuous trait. The proposed numerical methods can capture the satisfying long-time behavior of solutions. The main contents are as follows:(1) A finite volume type approximation gives semi-discrete and fully discrete schemes for a direct competitive model. The existence and uniqueness of discrete ESD(Evolutionary Stable Distribution) are proved through the equivalence between the problem of finding the discrete ESD and solving the associated quadratic programming problem. Then, the e?cient computation of the ESD can be carried out by any well established quadratic programming solver. Semi-discrete and fully discrete schemes are shown to satisfy the positivity and entropy dissipation inequalities under appropriate conditions on discrete coe?cients and time steps. An alternative algorithm is presented to capture the global ESD for nonnegative initial data, which is made possible due to the mutation mechanism built in the modified scheme. All the results for one dimensional case are extended to those for multi-dimensional case. Both one and twodimensional numerical results are provided to demonstrate both the accuracy and the entropy satisfying property, and numerical results underline the e?ciency to capture the large time asymptotic behavior of numerical solutions.(2) For proposed semi-discrete and fully discrete scheme, various time-asymptotic convergence rates of numerical solutions towards ESD are established. For some special ESD satisfying a strict sign condition, the thesis starts with a symmetrization of the system with weight depending on the strict ESD, and then obtain exponential decay of the perturbations using a Lyapunov functional approach, subject to a parameter tuned to allow for the largest possible initial perturbations. Finally the optimal convergence rate is obtained by a refined estimate. Using the proof idea, the thesis establishes the exponential convergence of numerical solutions towards such an ESD. For a general ESD, the thesis establishes the dissipation inequality of relative entropy and shows the decreasing property of the dissipation rate. These together ensure the algebraic convergence rate of numerical solutions towards the general ESD, which is consistent with the known result for the continuous model.(3) This thesis designs, analyzes and numerically validates energy dissipating finite volume schemes for a competition–mutation equation with a gradient flow structure. The discrete positive steady state is proven to be the same as the minimizer of the discrete energy function, so the discrete positive steady state uniquely exists. Thus,the positive steady state can also be produced by a well established nonlinear programming solver. Both semi-discrete and fully discrete schemes are demonstrated to satisfy positivity of numerical solutions and energy dissipation. These ensure that the positive steady state is asymptotically stable. A series of numerical tests is provided to demonstrate both accuracy and energy dissipation properties of numerical schemes. The large time behavior of numerical solutions is also shown.