Comparison of high-order accurate schemes for solving the nonlinear viscous burgers equation

In this paper, a comparison between higher order schemes has been performed in terms of numerical accuracy. Four finite difference schemes, the explicit fourth-order compact Pade scheme, the implicit fourth-order Pade scheme, flowfield dependent variation (FDV) method and high order compact flowfie ld dependent variation (HOC-FDV) scheme are tes ted. The FDV scheme is used for time disc retization and the fourth-order compact Pade scheme is used for spatial derivatives. The solution procedures consist of a number of tri-diagonal matrix operations and produce an efficient solver. The comparisons are performed using one dimensional nonlinear viscous Burgers equation to demonstrate the accuracy and the convergence characteristics of the high-resolution schemes. The numerical results show that HOC-FDV is highly accurate in comparison with analytical and with other higher order schemes

On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations

The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical stability and convergence are presented. Comparisons are made between the present schemes in terms of accuracy and computational efficiency for solving problems with severe internal and boundary gradients. The present study shows that the fourth-order compact ADI scheme is stable and efficient

A High- order Compact Finite Difference Solver for the Two-Dimensional Euler And Navier-Stroker Equations

The objective of this study was to develop a high-order compact (HOC) finite difference solver for the two-dimensional Euler and Navier-Stokes equations. Before developing the solver, a detailed investigation was conducted for assessing the performance of the basic fourth-order compact central discretization schemes that are known as Hermitian or Pade schemes. Exact solutions of simple scalar model problems, including the one-dimensional viscous Burgers equation and two dimensional convection-diffusion equation were used to quantitatively establish the spatial convergence rate of these schemes. Examples of two-dimensional incompressible flow including the driven cavity and the flow past a backward facing step were used for qualitatively evaluating the accuracy of the discretizations. Resolution properties of the HOC and conventional schemes were demonstrated through Fourier analysis. Stability criteria for explicit integration of the convection-diffusion equation were derived using the on-Neumann method and validated.Due to aliasing errors associated with the central HOC schemes investigated. these were only used for the discretization of the viscous terms of the Navier-Stokes equations in developing the aimed solver. Dealiasing HOC methods were developed for the discretization of the Euler equations and the convective terms of the Navier- Stokes equations. Spatial discretization of the Euler equations was based on flux-vector splitting. A fifth-order compact upwind method with consistent boundary closures was developed for the Euler equations. Shock-capturing properties of the method were based on the idea of total variation diminishing (TVD). The accuracy, stability and shock capturing issues of the developed method were investigated through the solution of one-dimensional scalar conservation laws. Discretization of the convective flux terms of the Navier-Stokes equations was based on a hybrid flux-vector splitting, known as the advection upstream splitting method (AUSM), which combines the accuracy of flux-difference splitting and the robustness of flux-vector splitting. High-order accurate approximation to the derivatives was obtained by a fourth-order cell-centered compact scheme. The midpoint values of the staggered mesh were constructed using a fourth-order MUSCL (monotone upstream-centered scheme for conservation law) polynomial. Two temporal discretization methods were built into the developed solver. Explicit integration was performed using a multistage strong stability preserving (SSP) Runge-Kutta method for unsteady time-accurate flow problems. For steady state flows an implicit method using the lower-upper (LU) factorization scheme with local time stepping convergence accelerator was employed. An advanced two-equation turbulence model, known as k-o shear-stress-transport (SST), model has also been incorporated in the solver for computing turbulent flows. A wide variety of test problems in unsteady and steady state were solved to demonstrate the accuracy, robustness and the capability to preserve positivity of the developed solver. Although the main solver was developed for two-dimensional problems, a one-dimensional version of it has been used to solve some interesting and challenging one-dimensional test problems as well. The test problems considered contain various types of discontinuities such as shock waves, rarefaction waves and contact surfaces and complicated wave interaction phenomena. Quantitative and qualitative comparisons with exact solutions, other numerical results or experimental data, whichever is available, are presented. The tests and comparisons conducted have shown that the developed HOC methods and the solver are high-order accurate and reliable as an application CFD code for two-dimensional compressible flows and conducting further research. A number of avenues for further research are identified and proposed for future extension and improvement of the solve

Proper orthogonal decomposition closure models for fluid flows: Burgers equation

This paper puts forth several closure models for the proper orthogonal decomposition (POD) reduced order modeling of fluid flows. These new closure models, together with other standard closure models, are investigated in the numerical simulation of the Burgers equation. This simplified setting represents just the first step in the investigation of the new closure models. It allows a thorough assessment of the performance of the new models, including a parameter sensitivity study. Two challenging test problems displaying moving shock waves are chosen in the numerical investigation. The closure models and a standard Galerkin POD reduced order model are benchmarked against the fine resolution numerical simulation. Both numerical accuracy and computational efficiency are used to assess the performance of the models