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A new L-stable Simpson-type rule for the diffusion equation

The well-known Simpson rule is an optimal two-step fourth-order method which is unconditionally unstable. In the present paper we describe a new L-stable version of the method. A suitable combination of the arithmetic average approximation with the explicit backward Euler formula provides a third-order approximation at the midpoint which, when plugged into the Simpson rule, gives a third-order L-stable scheme. The L-stable Simpson-type rule (LSIMP3) obtained is then employed to derive a third-order time integration scheme for the diffusion equation. Numerical illustrations are provided to compare the performance of the new LSIMP3 scheme with the Crank–Nicolson scheme. While the Crank–Nicolson scheme can produce unacceptable oscillations in the computed solution, the present LSIMP3 scheme can provide both stable and accurate approximations.