Wise-Use of Sediment for River Restoration: Numerical Approach via HJBQVI

Abstract

A stochastic differential game for cost-effective restoration of river environment based on sediment wise-use, an urgent environmental issue, is formulated. River restoration here means extermination of harmful algae in dam downstream. The algae population has weak tolerance against turbid river water flow, which is why the sediment is focused on in this paper. Finding the optimal strategy of the sediment transport reduces to solving a spatio-temporally 4-D Hamilton-Jacobi-Bellman Quasi-Variational Inequality (HJBQVI): a degenerate nonlinear and nonlocal parabolic problem. Solving the HJBQVI is carried out with a specialized finite difference scheme based on an exponentially-fitted discretization with penalization, which generates stable numerical solutions. An algorithm for solving the discretized HJBQVI without resorting to the conventional iterative matrix inversion methods is then presented. The HJBQVI is applied to a real problem in a Japanese river where local fishery cooperatives and local government have been continuing to debate the way of using some stored sediment in a diversion channel for flood mitigation. Our computational results indicate when and how much amount of the sediment should be applied to the river restoration, which can be useful for their decision-making.