Numerical Methods For The Forward-backward Heat Equation

In this paper, we study the numerical methods of the forward-backward heat equation in one dimension and two dimension, including the difference methods, the error estimate and numerical solving.The difference method is one of the approximate methods for the partial differential equations. In [22]. the author give an difference scheme for the forwardbackward heat equation, that is, they used the forward and backward difference scheme on subdomain respectively and second order difference scheme on the interface line. In our paper, we use second difference scheme on the interface with coarse mesh.Let u be the exact solution of problem (2.1) and z_i^j be the solution of (2.23)-(2.25), then the error E_i^j = u(ih,jÏ„) - z_i^j satisfies:Theorem 2.2 Suppose that (?) and (?) are bounded by constantC₀ on (?), the closure ofΩ. Thenwhich (?).Because our scheme is implicit for the forward-backward heat equation, then we discuss the iterative method based on the domain decomposition method to the difference equations. We derive the convergent rate for our iterative algorithm .Theorem 2.3 Letφ^j,k(1≤j≤N - 1, k = 0,1,…) be the solutions of equations(2.37)-(2.39), z₀^j(1≤j≤N - 1) be the solution of equations (2.23)-(2.25). ThenSoφ^j,k converge to z₀^j with the rate 1 - H as k→∞. It is better than 1 - h in [22]. Furthermore, for the two dimension forwardbackward heat equation we do it.Theorem 3.1 Suppose that (?) are bounded byconstant C₀ on (?), the closure ofΩ. ThenwhichIn addition,Theorem 3.2 For 1≤j≤M - 1,1≤k≤N - 1,p = 0,1,…, we haveExplicit schemes are often naturally parallel and also easy to implement, but they usually require small time steps because of stability constraints. Implicit schemes are necessary for finding steady state solutions or computing slowly unsteady problems where one needs to march with large time steps; however, implicit schemes are not inherently parallel because at each time step essentially an problem needs to be solved. However, Saulyev scheme is unconditional stable and explicit for the usual heat equation. Then in this, we apply it to the forwardbackward heat equation. Based on this, we construct the grouped scheme. They are all implicit in our problem, so we consider the iterative algorithm and derive the following result.Theorem 2.4 Whenα∈(0,1), the iterative algorithm is convergent.