High Order Finite Difference Methods For Two Types Of Elliptic Pdes Of Boundary Value Problems

Finite difference methods are important numerical tools in solving differential equations,in which derivatives are approximated by finite difference approximations.In this thesis,two types of elliptic partial differential equations including steady convectiondiffusion equations with variable coefficients and elliptic interface problem are solved numerically by finite difference methods.Convection diffusion equations are important partial differential equations and mathematical models for many applications including mass,energy and heat transfer process.In this thesis,an integrating factor idea is proposed to transform a variable coefficients steady convection diffusion equation to a self-adjoint variable coefficient diffusion equation.Then,two different fourth order numerical methods are employed to solve the variable coefficients diffusion equation.One is the fourth order accurate method obtained by the Richardson extrapolation,and the other is a fourth-order accurate compact scheme.Finally,numerical results are presented to verify the convergence of the two methods.Many practical applications can be modeled as interface problems in which the partial differential equations involve discontinuous coefficients and solutions,and/or singular source terms.It is important and challenging to solve interface problems both theoretically and numerically.It is well-known that the Immersed Interface Method(?M)is a second order accurate method for interface problems.However the accuracy of the first order derivatives,or gradients for short,is not so clear and is often assumed to be first order accurate.In this thesis,new strategies based on the ?M are proposed for elliptic interface problems to compute the gradient of the solution at both regular and irregular grid points,and at the interface from each side of the interface.Second order convergence for onedimensional(1D),or nearly second order(except a factor of |logh|)convergence for two-dimensional(2D)elliptic interface problems are presented.The computed gradient is obtained with almost no extra cost,and has been explained in intuition and verified by non-trivial numerical tests.Numerical examples in one,two dimensions,radial and axis-symmetric cases in polar and spherical coordinates are presented to validate the numerical methods and analysis.