Unconditionally stable high accuracy compact difference schemes for multi-space dimensional vibration problems with simply supported boundary conditions

Abstract

• We have derived 2-level implicit methods for 4th order vibration problems. • The methods are compact and unconditionally stable. • We do not require to discretize the boundary conditions. • We require only a tri-diagonal solver to compute the proposed numerical methods. • We have solved 09 practical examples arising from physics. The Euler–Bernoulli beam equation is a fourth order parabolic partial differential equation governing the transverse vibrations of a long and slender beam and is thus of interest in various engineering applications. In this study, we propose new two-level implicit difference formulas for the solution of vibration problem in one, two and three space dimensions subject to appropriate initial and boundary conditions. The proposed methods are fourth order accurate in space and second order accurate in time and are based upon a single compact stencil. The boundary conditions are incorporated in a natural way without any discretization or introduction of fictitious nodes. The derived methods are shown to be unconditionally stable for model linear problems. Some physical examples and their numerical results are given to illustrate the accuracy of the proposed methods. The test problems confirm that the computed solutions are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.