Weighting factors for single-step trapezoidal method

The uses of a weighting factor along with a time step in a single-step trapezoidal method to solve a first-order parabolic system have been systematically studied. The weighting factors are used in two main types: constants and variables. The most commonly used constant weighting factors can be defined by the ratio of the Fibonacci sequence. Among them, the optimal weighting factor is 0.618, resulting in a balance between the overall accuracy and efficiency. With the finite element formulation, the space and time dimensions can be discretized separately. For the time discretization only, there exists a zero-error dimensionless time step if a weighting factor is within the range of 0.5-1.0. By taking advantage of the zero-error condition, the weighting factor can be correlated with a time step. The influence of spatial dimensions is lumped into a nonzero eigenvalue of the system. Through validity tests of two benchmark linear problems, the variable weighting factor for a single-step trapezoidal method is shown to be accurate, efficient, and stable. The relevant features have been captured.