High-order Accurate Finite Difference Scheme Research Articles

A high-order accurate finite difference scheme for the KdV equation with time-periodic boundary forcing

In this article, we present an accurate finite difference scheme for solving the KdV equation with time-periodic forcing at boundary. We first employ a variable transformation to change the time-periodic boundary to a homogeneous boundary. We then develop a three-level fourth-order accurate finite difference scheme for solving the transformed KdV problem. The stability, existence and uniqueness of the numerical solution are analyzed. Numerical examples are provided to confirm the accuracy of the scheme, and the effectiveness for time-periodic simulation.

A nonlinear filter for high order discontinuous Galerkin discretizations with discontinuity resolution within the cell

The nonlinear filter introduced by Yee et al. (1999) [27] and extensively used in the development of low dissipative well-balanced high order accurate finite-difference schemes is adapted to the finite element context of discontinuous Galerkin (DG) discretizations. The filter operator is constructed in the canonical computational domain for the standard cubical element where it is applied to the computed conservative variables in a direction per direction basis. Filtering becomes possible for all element types in unstructured meshes using collapsed coordinate transformations. The performance of the proposed nonlinear filter for DG discretizations is demonstrated and evaluated for different orders of expansions for one-dimensional and multidimensional problems with exact solutions. It is shown that for higher order discretizations discontinuity resolution within the cell is achieved and the design order of accuracy is preserved. The filter is applied for a number of standard inviscid flow test problems including strong shocks interactions to demonstrate that the proposed dissipative mechanism for DG discretizations yields superior results compared to the results obtained with the total variation bounded (TVB) limiter and high-order hierarchical limiting. The proposed approach is suitable for p-adaptivity in order to locally enhance resolution of three-dimensional flow simulations that include discontinuities and complex flow features.

Parametric numerical study of electrokinetic instability in cross-shaped microchannels

A multiblock in-house solver based on high-order accurate finite-difference schemes is developed to simulate electrokinetic instability (EKI) phenomena in realistic cross-shaped microchannels. Parametric numerical studies are systematically carried out to investigate the effects of different parameters on EKI including the conductivity ratio, the electric field strength, the channel depth, and the electric field ratios. Simulation results show that the solver has the capability to capture the threshold electric field for the onset of instabilities and it offers a good prediction for the critical features of EKI compared to the experiments presented in the literature. In addition, from the general parametric studies several useful guidelines are explored. Such guidelines can be applied to enhance the mixing efficiency in electrokinetic microdevices.

High-fidelity numerical simulation of solitons in the nerve axon

High-order accurate finite difference schemes are derived for a non-linear soliton model of nerve signal propagation in axons. Two types of well-posed boundary conditions are analysed. The boundary closures are based on the summation-by-parts (SBP) framework and the boundary conditions are imposed using a penalty (SAT) technique, to guarantee linear stability. The resulting SBP-SAT approximation is time-integrated with an explicit finite difference method. The accuracy and stability properties of the newly derived finite difference approximations are demonstrated for an analytic soliton solution.

Temporal and spatial evolution characteristics of disturbance wave in a hypersonic boundary layer due to single-frequency entropy disturbance.

By using a high-order accurate finite difference scheme, direct numerical simulation of hypersonic flow over an 8° half-wedge-angle blunt wedge under freestream single-frequency entropy disturbance is conducted; the generation and the temporal and spatial nonlinear evolution of boundary layer disturbance waves are investigated. Results show that, under the freestream single-frequency entropy disturbance, the entropy state of boundary layer is changed sharply and the disturbance waves within a certain frequency range are induced in the boundary layer. Furthermore, the amplitudes of disturbance waves in the period phase are larger than that in the response phase and ablation phase and the frequency range in the boundary layer in the period phase is narrower than that in these two phases. In addition, the mode competition, dominant mode transformation, and disturbance energy transfer exist among different modes both in temporal and in spatial evolution. The mode competition changes the characteristics of nonlinear evolution of the unstable waves in the boundary layer. The development of the most unstable mode along streamwise relies more on the motivation of disturbance waves in the upstream than that of other modes on this motivation.

High-order accurate computations for unsteady aerodynamics

A high-order accurate finite difference scheme is used to perform numerical studies on the benefit of high-order methods. The main advantage of the present technique is the possibility to prove stability for the linearized Euler equations on a multi-block domain, including the boundary conditions. The result is a robust high-order scheme for realistic applications. Convergence studies are presented, verifying design order of accuracy and the superior efficiency of high-order methods for applications dominated by wave propagation. Furthermore, numerical computations of a more complex problem, a vortex–airfoil interaction, show that high-order methods are necessary to capture the significant flow features for transient problems and realistic grid resolutions. This methodology is easy to parallelize due to the multi-block capability. Indeed, we show that the speedup of our numerical method scales almost linearly with the number of processors.

Novel Defect-correction High-order, in Space and Time, Accurate Schemes for Parabolic Singularly Perturbed Convection-diffusion Problems

AbstractNew high-order accurate finite difference schemes based on defect correction are considered for an initial boundary-value problem on an interval for singularly perturbed parabolic PDEs with convection; the highest space derivative in the equation is multiplied by the perturbation parameter ε. Solutions of the well-known classical numerical schemes for such problems do not converge ε-uniformly (the errors of such schemes depend on the value of the parameter ε and are comparable with the solution itself for small values of ε). The convergence order of the existing ε-uniformly convergent schemes does not exceed 1 in space and time. In this paper, using a defect correction technique, we construct a special difference scheme that converges ε-uniformly with the second (up to a logarithmic factor) order of accuracy with respect to x and with the second order of accuracy and higher with respect to t.