An adaptive WENO algorithm for one-dimensional convection-dominated partial differential equations

Abstract

In this work, a versatile numerical algorithm was proposed for solving one-dimensional nonlinear convection-dominated partial differential equations as these found in the simulation of diverse chemical engineering applications. The proposed algorithm uses a fully adaptive scheme, in which both the grid spacing and the time discretization are dynamically adjusted. It uses a finite volume discretization with a variable number of grid cells and an explicit time integration with time-step control. The high-order Weighted Essentially Non-Oscillatory (WENO) scheme on non-uniform grid was combined with a grid refinement technique based on the equitable distribution principle and a spatial regularization procedure. The time-stepping procedure as well as some implementation issues are minutely discussed. The capability and efficiency of the new algorithm was demonstrated through the numerical simulation of five chemical engineering applications: the Viscous Burgers Equation, a Chromatographic column, the Buckley-Leverett Equation, and two examples of reaction-convection systems. For all analyzed cases, it was verified that the number of grid cells required to capture steep gradients can be greatly reduced with the proposed grid adaptation scheme.