Least‐squares schemes for time integration of thermal problems

The paper proposes a generalized least‐squares time finite element procedure for analysis of transient field problems. It leads to a very general four parameter and two time level family of schemes. The free parameters are optimized in terms of accuracy, stability and oscillatory behaviour of the computational algorithm. In the case of linear problems an unconditionally stable scheme of fourth‐order accuracy is obtained. Special choices of the free parameters also give a number of well‐known time‐integration schemes, such as the Zienkiewicz–Lewis third‐order algorithm, the Goodrich scheme and standard schemes such as the Crank–Nicolson and backward fully implicit schemes. The main disadvantage of the higher order algorithm is a large bandwidth of the system matrix owing to the matrix terms of the form KTK. This drawback is reduced for special choices of the parameters by forming a product formula. Somewhat more computation is required than for popular schemes such as the generalized midpoint rule, depending on the prescribed variables. Numerical comparisons are made between the proposed higher order schemes and standard schemes.