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In this paper, three novel truly-explicit time-marching techniques are proposed for the solution of hyperbolic models. In these techniques, locally computed time-integration parameters are applied, which are defined taking into account the properties of the spatially/temporally discretised model and the evolution of the computed responses. These adaptive time integrators allow introducing enhanced numerical damping into the analysis, reducing spurious non-physical oscillations that occur due to the excitation of spatially unresolved modes. The proposed adaptive techniques are formulated based on second-, third-, and fourth-order accurate time-marching approaches, providing solution algorithms with different complexities. At the end of the paper, numerical results are presented, illustrating the good performance of the proposed procedures.

In this paper, η-order accurate truly-explicit time-marching procedures are proposed for dynamic analyses. In these new techniques, just η–1 stiffness matrix–vector operations are required for solution, per time-step, providing more efficient approaches than standard high-order time-marching procedures, which demand η matrix–vector operations per time-step. In addition, as illustrated along this work, the novel formulations are also more accurate than equivalent standard techniques, providing very effective time-domain solution procedures. Three truly-explicit time-marching procedures are discussed here, describing second-, third- and fourth-order accurate methodologies.

In this paper, a novel time-marching procedure is proposed for dynamic analysis. The proposed new technique is fourth-order accurate for undamped models; third-order accurate for numerically damped models; and second-order accurate for physically damped models. The method is very straightforward to implement and to apply, allowing providing higher-order accurate time-domain solutions taking into account basically the same computational effort of standard single-step first- or second-order accurate time-marching procedures. Numerical examples are presented at the end of the paper, illustrating the excellent accuracy of the novel approach.

In this article, a meshless spectral radial point interpolation method is proposed for the numerical solutions of a class of time-fractional advection–diffusion–reaction equations. The current approach utilizes meshless shape functions, having Kronecker delta function property, for approximation of spatial operators. Forward difference along with quadrature formula is used for tempered fractional derivative approximation in the framework of implicit time marching scheme. Assessment of the proposed method is made by applying to various concrete test problems having constant and variable coefficients. Approximation and function reproduction quality are measured via $${E}_{\infty }$$ , $${E}_{2}$$ and $${E}_{\text {rms}}$$ error norms. Comparison of simulated results is also made with available exact solutions as well as earlier reported works. Stability analysis of the proposed method is thoroughly discussed and computationally affirmed.