High accuracy computing methods : fluid flows and wave phenomena - Sengupta, Tapan Kumar, 1955- (Cambridge : Cambridge University Press, 2013)
This book presents topics in a single source format using unified spectral theory of computing. With developments of DNS and LES, practitioners are rediscovering waves as important in fluid flows, and capturing these numerically is central to high accuracy computing. Analysis of waves and its use in numerical methods in propagating energy at the right velocity (dispersion effects) and with right amplitude (dissipation) are essential. Most industrial codes using Reynolds-averaged Navier-Stokes equations with turbulence models cannot conceive of capturing waves. The new themes covered in this book are: Correct error propagation analysis Practical compact schemes and global analysis tool Aliasing error and its alleviation Spurious upstream propagating q-waves Explanation of the Gibbs phenomenon New 1D and 2D filters for LES/DNS without SGS modelling Anisotropic skewed wave propagation Development and analysis of dispersion relation preservation (DRP) schemes Flow instabilities and wave propagation phenomena
10 Spectral Analysis of Numerical Schemes and Aliasing Error -- 10.1.Introduction -- 10.2.Spatial Discretization of First Derivatives -- 10.2.1.Second order central differencing (CD2) scheme -- 10.3.Discrete Computing and Nyquist Criterion -- 10.4.Spectral Accuracy of Differentiation -- 10.5.Spectral Analysis of Fourth Order Central Difference Scheme -- 10.6.Role of Upwinding -- 10.6.1.First order upwind scheme (UD1) -- 10.6.2.Third order upwind scheme (UD3) -- 10.7.Numerical Stability and Concept of Feedback -- 10.8.Spectral Stability Analysis -- 10.9.High Accuracy Schemes for Spatial Derivatives -- 10.10.Temporal Discretization Schemes -- 10.10.1.Euler time integration scheme -- 10.10.2.Four-stage Runge-Kutta (RK4) method -- 10.11.Multi-Time Level Discretization Schemes -- 10.11.1.Mid-point leapfrog scheme -- 10.11.2.Second order Adams-Bashforth scheme -- 10.12.Aliasing Error -- 10.12.1.Why aliasing error is important? -- 10.12.2.Estimation of aliased component -- 10.13.Numerical Estimates of Aliasing Error -- 10.14.Controlling Aliasing Error -- 10.14.1.Aliasing removal by zero padding -- 10.14.2.Aliasing removal by phase shifts and grid-staggering -- ch. 11 Higher Accuracy Methods -- 11.1.Introduction -- 11.2.The General Compact Schemes -- 11.2.1.Approximating first derivatives by central scheme -- 11.3.Method for Solving Periodic Tridiagonal Matrix Equation -- 11.4.An Example of a Sixth Order Scheme -- 11.5.Order of Approximation versus Resolution -- 11.6.Optimization Problem Associated with Discrete Evaluation of First Derivatives -- 11.7.An Optimized Compact Scheme For First Derivative by Grid Search Method -- 11.8.Upwind Compact Schemes -- 11.9.Compact Schemes with Improved Numerical Properties -- 11.9.1.OUCS1 scheme -- 11.9.2.OUCS2 scheme -- 11.9.3.OUCS3 scheme -- 11.9.4.OUCS4 scheme -- 11.10.Approximating Second Derivatives -- 11.11.Optimization Problem for Evaluation of the Second Derivatives -- 11.12.Solution of One-Dimensional Convection Equation -- 11.13.Symmetrized Compact Difference Schemes -- 11.13.1.High accuracy symmetrized compact scheme -- 11.13.2.Solving bidirectional wave equation -- 11.13.3.Transitional channel flow -- 11.13.3.1.Establishment of equilibrium flow -- 11.13.3.2.Receptivity of channel flow to convecting single viscous vortex -- 11.13.4.Transitional channel flow created by vortex street -- 11.14.Combined Compact Difference (CCD) Schemes -- 11.14.1.A new combined compact difference (NCCD) scheme -- 11.14.2.Solving the Stommel Ocean Model problem -- 11.14.3.Operational aspects of the CCD schemes -- 11.14.4.Calibrating NCCD method to solve Navier-Stokes equation for 2D lid-driven cavity problem -- 11.15.Diffusion Discretization and Dealiasing Properties of Compact Schemes -- 11.15.1.Dynamics and aliasing in square LDC problem --
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Keywords: Fluid dynamics -- Data processing; Wave mechanics -- Data processing; Spectrum analysis -- Data processing