Finite Volume WENO Schemes And Applications

High order accurate weighted essentially non-oscillatory (WENO) schemes, whichare constructed based on the successful essentially non-oscillatory (ENO) schemes, arerelativelynewbuthavegainedrapidpopularityinnumericalsolutionsofhyperbolicpar-tial differential equations and other convection dominated problems. The main advan-tage of such schemes is their capability to achieve arbitrarily high order formal accuracyinsmoothregionswhilemaintainingstable,non-oscillatoryandsharpdiscontinuitytran-sitions. The WENO schemes can be designed in the finite difference or finite volumeframework. The finite volume scheme can be defined on arbitrary Cartesian meshes,even those with abrupt changes in mesh sizes, without affecting their conservation, ac-curacy and stability, in contrast to high order conservative finite difference schemeswhich can only be defined on smooth meshes. Finite volume schemes are also easierto implement in an adaptive mesh environment, for example in the AMR type schemes.This is the main reason that high order finite volume schemes are still commonly usedin practice, even though high order finite difference schemes are much less expensivein multi-dimensions in uniform or smooth Cartesian meshes. In this thesis, we will givediscussions on finite volume WENO schemes with application.We will first develop ahigh order positivity-preserving finite volume WENO schemefor solving a hierarchical size-structured population model with nonlinear growth, mor-tality and reproduction rates. We carefully treat the technical complications in boundaryconditions and global integration terms to ensure high order accuracy and positivity-preservingproperty. ComparingwiththeprevioushighorderdifferenceWENOschemefor this model, the positivity-preserving finite volume WENO scheme has a comparablecomputational cost and accuracy, with the added advantage in positivity-preserving andL~1 stability. Numerical examples including the one for the evolution of the populationof Gambusia affinis, are presented to illustrate the good performance of the scheme.Then we consider two commonly used classes offinite volume weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian meshes. We comparethem in terms of accuracy, performance for smooth and shocked solutions, and effi-ciency in CPU timing. For linear systems both schemes are high order accurate, how-ever for nonlinear systems, analysis and numerical simulation results verify that one ofthem (Class A) is only second order accurate, while the other (Class B) is high orderaccurate. The WENO scheme in Class A is easier to implement and costs less than thatin Class B. Numerical experiments indicate that the resolution for shocked problems isoften comparable for schemes in both classes for the same building blocks and meshes,despite of the difference in their formal order of accuracy. The results in this paper maygivesomeguidanceintheapplicationofhighorderfinitevolumeschemesforsimulatingshocked flows.