روش گالركين ناپيوسته تركيب شده براي معادلات Kdv مرتبه سوم و پنجم

چكيده انگليسي :

A hybridized discontinuous Galerkin method for third and fth order KdV equations Shima Baharloui s baharloui@math iut ac ir 2017 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac ir 2010 MSC 65M60 65N30 Keywords Hybrid discontinuous Galerkin methods Third and Fifth order KdV equations Abstract A class of nite element methods is discontinuous Galerkin DG methods that uses discontinuouspiecewise polynomial space for the numerical solution and the test functions In this thesis aftergathering some prerequisites about the Sobolev space its inner product Hibert space and projectionoperators DG methods are explained for a simple equation Because of discontinuity DG methodshave certain exibility and advantages As we know it is not suitable to solve some equations withhigher order spatial derivatives with DG methods So a new class of DG which is termed local DG LDG has been introduced to solve a convection di usion equation in 1998 Its idea is to rewrite ahigher order partial di erential equation such as Korteweg de Vries KdV equation into a rst ordersystem by introducing auxiliary variables and then solve it using a DG method In fact auxiliaryvariables are introduced to approximate derivatives of the solution A crucial part of such methods forthe success is the reasonable choice of interface numerical uxes which guarantee method s stabilityand local solvability of all of the auxiliary variables It must be pointed out that since the auxiliaryvariables can be locally eliminated LDG method can keep the exibility of the DG method After these what investigated is a hybridized discontinuous Galerkin HDG method for solving thethird and fth order KdV type equations which is the main part of our work HDG methods aredi erent from the previous DG methods These methods were initially developed in order to addressthe large number of degrees of freedom that more standard DG methods de ne for the steady stateproblems More precisely a discontinuous basis in DG methods cause discontinuous solutions alongelements faces So we have multivalued function evaluations at inner element uxes This increases thedegree of freedom in contrast to a continuous basis where function evaluations are single valued alongelements faces In this method at rst it must be expressed the auxiliary variables in order to formthe rst order system of equations and then introduced numerical uxes inside each element in termsof the numerical traces and stabilization parameters which have an important role in stabilizing themethod Related traces are assumed to be single valued on each face Next we impose conservationof the numerical uxes via two for third order KdV equations or four for fth order KdV equations extra sets of equations Using these global ux conservation conditions a system of linear nonlinearof equations will be established which can be solved using a suitable solver e g Newton Raphsonfor the nonlinear systems It should be noted that time variable is discretized by using a suitabledi erence formulae such as the backward Euler method Stability analysis of the HDG method isinvestigated by extracting a cell entropy inequality Some numerical examples are tested in order toshow that for a mesh with k th order elements approximate solution and its derivatives have optimalconvergence at order k 1 which is comparable with or even superior to the previous methods

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روش گالركين ناپيوسته تركيب شده براي معادلات Kdv مرتبه سوم و پنجم

فضاي سوبولف , شار عددي , معادله گرما , Kdv