Variable Coefficients and Nonlinear Problems

This chapter discusses the conservation and stability of difference schemes for linear and nonlinear model equations with variable coefficients. For the variable coefficients in linear partial differential equations, if the coefficients of the equation are smooth functions, similar methods to the constant coefficient case can be used to construct difference schemes. The stability and convergence can be analyzed using the frozen coefficient method. For the nonlinear partial differential equations, stability analysis and error estimation for nonlinear problems are more complex than linear problems. At the end of this chapter, the conservative difference scheme is discussed and analyzed using the controlling volume method based on the physical conservation law.