Numerical Methods With Preservation Of Physical Properties For The Reaction Diffusion Equations

Nowadays,scientific computation has been acted as the third important scientific research way with the fast development of society and application of computer science,which is of great importance as same as the theory and experiment.In many cases for the constrain of the unexpected factor and experiment equipment,computational simulation may probably derive the expected results at the lower cost than that in the theorem and experiment,which makes scientific computational stand high in the current scientific field.The reaction diffusion equation is a kind of important partial differential equations,illustrated in their theoretical analysis and numerous applications ranging from physics,biology,to materials and social science.As we know,the reaction diffusion equations satisfy some properties such as maximum principle,comparison principle,existence of invariant sets,and energy decay.These properties represent important features and are also essential for mathematical analysis and numerical simulations.Thus,it is necessary to study and construct the numerical method to satisfy these properties.Main context in this thesis is as follows:First is the integrating factor method on the maximum bound principle of the semilinear parabolic equation.As we know,Strong stability preserving integrating fac-tor Runge-Kutta methods are an attractive alternative to traditional strong stability preserving Runge-Kutta methods when evolving in time that have the linear compo-nent alleviate restrictive time-steps.Where maximum bound principle,weaker than the strong stability preserving property,is needed,it has been shown that integrating factor Runge-Kutta methods have the restrictive time-steps coming from the nonlin-ear component and are thus not efficient.In this work we show that it is possible to define explicit integrating factor Runge-Kutta methods that preserve the desired max-imum bound principle by introducing the stabilizing coefficient.We define sufficient conditions for explicit stabilizing integrating factor Runge-Kutta methods to preserve maximum bound principle,namely,that they are based on explicit Runge-Kutta meth-ods with strictly increasing abscissas.We find such methods of up to three order and up to six stages,analyze their maximum bound principle.We test these methods to demonstrate their convergence and to show that the strictly increasing abscissa condi-tion is needed in our test cases.Finally,some numerical experiments are carried out to show the maximum bound principle of all proposed methods.Second is the virtual element method on the maximum principle of the reaction diffusion equation on surface.The crucial feature of virtual element method is that it can be seen as the extension of the standard finite element method,i.e.the method can easily treat the polygon element rather than triangular element.The advantage of virtual element method has been explored on a variety of equations.However,it is a challenge to extend virtual element method to surface PDEs.Frittelli et al has proposed a surface virtual element method,which has a difficult in that the discretized polygonal elements are required to be flat,that is,that is,all vertexes of a element must be in a same plane.This method can well construct the virtual element space and obtain the good convergence analysis.However,it also causes a problem that the construction of flat element in general surfaces is not easy to implement.To overcome this difficulty,we design a Voronoi-based local tangential lifting virtual element method.This method combines the surface virtual element method with the local tangential lifting method in order to treat the non-flat Voronoi discretized surfaces.Non-flat polygon element is lifted to tangential plane where we can easily construct the local virtual element space.we derive the error bounds of discretized bilinear forms based on the standard H1 projector and L2 projector of lifted polygonal element in the tangential plane.Numerical experiments are provided to confirm the convergence results and show the efficiency of the proposed method.Using the mass lumped method,we can maintain the extreme-preserving properties of reaction diffusion equation.Finally is the radial basis function method on the energy dissipation of the Stokes equations on surfaces.One of the difficulties in numerically solving the incompressible Stokes equations is the couple of the velocity and pressure by the inf-sup condition.Thus many numerical methods were proposed to overcome the condition,among which mainly can be divided into two categories.One is to use projection methods,Broadly speaking,projection methods employ operator splitting,and use the pressure to project an intermediate velocity field to the space of incompressible or divergence-free velocity fields.However,they typically require specialized grids,and the careful selection of pressure and intermediate velocity boundary conditions to match the actual boundary conditions on the velocity field.The other numerical method is to use pressure Poisson equations,which is based on the Helmholtz decomposition.And from the pressure Pois-son equations,velocity can be a unique evolution variable with the pressure regarded as a implicit function.The couple of velocity and pressure is only in space,which can be seen as alleviation of the constrain of inf-sup condition.Based on this,we interpolate the non-divergence velocity in the equivalent equation generated by the incompressible Stokes equations using the surface Helmholtz decomposition.A rigorous analysis is given to point out the stability and convergency of the method.Numerical examples are presented demonstrating the efficiency of some model problems on more general surface and show that the discrete kinetic energy decays monotonically.