Finite difference schemes for the fourth‐order parabolic equations with different boundary value conditions

Abstract

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(𝜏 2 + h 2 ) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.

The compact difference approach for the fourth‐order parabolic equations with the first Neumann boundary value conditions

“…Example 1) H 1 semi-norm errors and convergence orders in space (𝜏 = 1/100,000). Lu et al[28] can also be observed. In addition, we can see that the fourth-order scheme (3.26)-(3.31) can reduce the storage and computational cost significantly.…”

“…The global convergence order of this scheme in the H 1 semi-norm was proved to be O(𝜏 2 + h 2 ) in Lu et al [28]. Denote…”

“…Meanwhile, the comparison with the difference scheme established in Lu et al [28] will be made. For completeness, the difference scheme for solving the problem (5.1)-(5.4) proposed in Lu et al [28] is listed here as…”

“…Table 3 shows the computational results of the difference scheme (3.26)-(3.31) and the scheme in Lu et al [28] under the optimal step size ratio, that is, taking 𝜏 = h 2 for the scheme (3.26)-(3.31) and 𝜏 = h for the latter one, from which the fourth-order numerical accuracy of the scheme (3.26)-(3.31) in space and the second-order accuracy of the scheme…”

“…where 𝑓 (x, t), 𝜑(x), 𝛼 i (t), 𝛽 i (t)(i = 1, 2) are all given functions and 𝜑 ′ (0) = 𝛼 1 (0), 𝜑 ′ (L) = 𝛽 1 (0), 𝜑 ′′ (0) = 𝛼 2 (0), 𝜑 ′′ (L) = 𝛽 2 (0). Lu et al [28] proposed a second-order difference scheme to solve this problem, where two quartic interpolation polynomials were built based on the given boundary conditions to obtain the numerical approximation of the fourth-order spatial derivative adjacent to the boundaries and the standard five-point central difference quotient was applied to approach the fourth-order spatial derivative at the interior mesh points. The second-order convergence of the resultant scheme in the discrete H 1 semi-norm was proved by the energy method.…”