One of the main concerns of Computational Fluid Dynamics (CFD) practitioners is to improve the accuracy and resolution of the computed flow field. The accuracy and resolution of the computations are affected a great deal by the numerical scheme for discretizing the governing equations of the flow. In compressible-flow computations the issues of accuracy and resolution are more complex due to the presence of features like shocks, slip-lines, contact discontinuities, and various types of wave interactions- such as shock reflection, shock-shock interaction, shock- expansion interaction etc.
Historically, in the beginning the method of computing compressible flows in- volving shocks was that of “Shock Fitting”, in which the states across the shock were connected using the Rankine-Hugoniot relations and the governing equations in the upstream and downstream zones of the shock were numerically solved at each time step. However, this approach has its own pitfalls in that the shock surface itself may be in motion relative to the network of points in space-time coordinates, and the differential equations as well as the boundary conditions are non-linear.
Moreover, the shape and location of the shock surface is not always known in advance, which are determined by the governing equations and the boundary con- ditions themselves [1]. Consequently, the shock fitting approach involves lengthy calculations that are mostly done with trial and error at each time step.
In the presence of shocks or contact discontinuities, the flow variables become
non-differentiable at the locations of the discontinuities. Clearly, discontinuous solutions do not satisfy the partial differential equation (PDE) in the classical sense at all points, since the space-derivatives of the variables are not defined at these locations. To overcome these difficulties, “weak solution” to the governing PDE can be sought. A weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. One approach to obtain weak solutions is the “vanishing-viscosity method”, where a diffusive term is introduced into the governing equations to obtain a unique-smooth solution and then the solution is sought at the limit when the coefficient of the diffusive term approaches zero. The addition of artificial viscosity results in smearing of the discontinuities. As a result of this the mathematical surfaces are replaced by thin layers where flow variables like pressure, velocity, density etc. change rapidly but smoothly. Addition of artificial viscosity and solving the governing equations in conservation form using a stable numerical scheme with properly posed initial and boundary conditions resulted in automatic capturing of the shock-shape and location during the space-time evolution. This new approach resulted to a shift of the computational paradigm from “Shock Fitting” to “Shock Capturing”.
The dynamics of inviscid-compressible flows is governed by the unsteady Euler equations of gas dynamics that are hyperbolic in nature. For hyperbolic con- servation laws, it is a well-established fact that the Forward-Time Central-Space (FTCS) method is unconditionally unstable. To impart numerical stability it is essential to add some numerical diffusion to the central dicretization of the space- derivatives associated with the inviscid fluxes. The major challenge in achieving high accuracy in compressible-flow computations is to ensure that a right amount of stabilizing diffusion is added in regions of sharp gradients to ensure stability, while in the rest of the computational domain the diffusion must be small enough not to spoil the solution accuracy. It has to be always borne in mind that too high numerical diffusion causes nonphysical smearing of the discontinuities.
To illustrate the point let us consider the one-dimensional (1D) linear advection
equation
∂u
∂t +c∂u
∂x = 0, (1.1)
where c = constant > 0 is the wave speed. Let equation 1.1 be discretized by using a first-order-forward difference in time and first-order-rearward difference in space. Then equation 1.1 can be represented by the following difference equation:
un+1j −unj
∆t +cunj −unj−1
∆x = 0, (1.2)
where unj and unj−1 represent the discrete solutions for u at locations xj and xj−1
at time t, respectively, ∆x is the grid-spacing, ∆t is the time-step, and un+1j is the discrete solution at xj at time t + ∆t. Equation 1.2 is the first-order- upwind discretization of equation 1.1. It can be shown that equation 1.2 is actually equivalent to solving another partial differential equation of the form [2]:
∂u
∂t +c∂u
∂x = c∆x
2 (1−ν)∂2u
∂x2 +c(∆x)2
6 3ν−2ν2 −1∂3u
∂x3 +O
(∆t)3,(∆t)2∆x,∆t(∆x)2,(∆x)3
, (1.3) where ν= c∆t∆x. The exact solution (free from any round-off error) of equation 1.2 can be viewed as the numerical solution of theoriginal partial differential equation 1.1 with an error given by the truncation error. However, from another perspective, the exact solution (free from any round-off error) of the difference equation 1.2 constitutes an exact solution (no round-off error) of a different partial differential equation, namely, equation 1.3 that is called the modified equation. The leading- order term on the right-hand side of equation 1.3 contains a term ∂∂x2u2 that has viscous-like effects. Though the original equation 1.1 was free from any viscous effects, the upwind discretization of the equation has artificially created viscous- like effects that are purely of numerical origin. The artificial viscosity or numerical diffusion results in smoothing of discontinuities in the marching procedure. Figure 1.1 shows a qualitative picture of the smearing of an initial discontinuity due to numerical diffusion in computing the 1D linear advection equation.
Fig. 1.1. Effect of numerical diffusion in computing the 1D linear wave equation with an initial discontinuity.