Numerical methods for the multiplicative partial differential equations

We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.

ERROR ESTIMATION FOR QUADRATURE FORMULAS BASED ON EQUALLY SPACED NODES

The error estimation for quadrature formulas based on equally spaced nodes is discussed in this paper. The error estimates use embedded formulas and they are obtained for Newton‐Cotes and Hermitian quadrature formulas. The coefficients of these formulas and error estimates are presented. The locally adaptive integration procedures implementing the truncation error estimation method proposed in this paper were developed in MATLAB and results of appropriate comparative tests are presented.

Variable step Rosenbrock algorithm for transient response of damped structures

A Rosenbrock algorithm with varying time step is adapted for transient analysis of damped second-order differential equations. The time step adjustment is based on an embedded local truncation error estimation formula. An interpolation formula can be used for intermediate output. The stepping formula is L-stable and the error estimation formula is bounded for large time steps. The Rosenbrock algorithm is compared with the Thomas—Gladwell STEP34 algorithm, which is found to be only conditionally stable. Numerical results are given for two linear examples: a stiff, linear, two-degree-of-freedom system and a non-proportionally damped plate.