An alternating direction generalized trapezoidal formula scheme for parabolic differential equations in two space dimensions

Abstract

A well-known ADI scheme for parabolic differential equations in two space dimensions is the Peaceman-Rachford scheme; this scheme employs the backward Euler formula for integration in time and is unconditionally stable. An ADI Crank -Nicolson scheme, which employs the classical trapezoidal formula for integration in time, is unconditionally unstable. We investigaan ADI implementation of the generalized trapezoidal formula GTF(α) for integration in time. The obtained ADI-GTF(α) scheme is unconditionally stable for all α ≥ 1; interestingly, ADIGTF(α) scheme includes the Peaceman-Rachford scheme for α→∞. Numerical experiments demonstrate that while the Peaceman-Rachford scheme can give quite pronounced unwanted oscillations in the computed solution, an ADI-GTF(α) scheme can provide a more stable and accurate approximation for the true solution.

Keywords:

• Parabolic Differential Equations
• Two Space Dimensions
• Adi Schemes
• Peaceman-Rachford Scheme
• Generalized Trapezoidal Formula Scheme
• Unconditional Stability

C.R. Category:

• G.1.8

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Notes

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