The Difference Method Simulation Of Parabolic Equation With Control Parameter

The parabolic equation with the control parameter is a class of parabolic inverse problems and is nonlinear. While determinating the solution of the problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. The article is devoted to the following parabolic initial and boundary problem with the control parameter:where Φ(x,t), f(x),g₀(t), g₁(t), E(t) are known functions, u(x,t) and p(t) are unknown functions.The article consists of two parts. In the first part, we construct a linearized Crank-Nicolson type difference scheme and a linearized compact difference scheme for the parabolic equation with control parameter. For the former, the accuracy of the discretization are two order both in time and in space. For the latter, the accuracy of the discretization is two order in time and four order in space. We analyze the solvability of the two difference schemes. We compare the numerical rusults of those two difference schemes with the four difference schemes in the reference [5]. The numerical results show that the two linearized difference schemes of the article improve the accuracy of the space and time direction and shorten computation time largely.In the second part, with a function transformation we transform the nonlinear problem to linear problem, then construct the difference scheme for the latter. We analyze the stability and convergence of the difference scheme with the maximum principle.