## 1 INTRODUCTION

High-order CFD methods have received considerable attention in the research community in the past two decades because of their potential in delivering higher accuracy with lower cost than low-order methods. Before we proceed any further, let us first clarify what we mean by order of accuracy and *high order*. Mathematically, a numerical method is said to be the *k*th order (or order *k*) if the solution error *e* is proportional to the mesh size *h* to the power *K*, that is, *e* ∝ *h*^{k}. In 2007, when the first author became the chair of the CFD Algorithm Discussion Group (CFDADG) in the American Institute of Aeronautics and Astronautics Fluid Dynamics Technical Committee (AIAA FDTC), a survey about the definition of *high order* was sent to members of the technical committee and other researchers outside the technical committee. Amazingly, we received a unanimous definition of high order: third order or higher. This is perhaps because nearly all production codes used in the aerospace community are first or second order accurate. We do understand that in certain communities, only spectral methods are considered high order.

Many types of high-order methods have been developed in the CFD community to deal with a diverse range of problems. At the extremes of the accuracy spectrum, one finds the spectral method [1] as the most accurate and a first-order scheme (e.g., the Godunov method [2]) as the least accurate. Many methods were developed for structured meshes, for example, [3-8]. Other methods were developed for unstructured meshes, for example, [6, 9-18]. For a review of such methods, see [19] and [20]. The purpose of the present paper is not to review high-order methods but to measure the performance of these methods as fairly as possible. In addition, we wish to dispel some myths or beliefs regarding high-order methods.

### Belief 1. High-order methods are expensive

This one is among the most widely held *belief* about high-order methods. The myth was perhaps generated when a CFD practitioner programmed a high-order method and found that obtaining a converged steady solution with the high-order method took much longer than with a low-order method *on a given mesh*. It is well known that a second-order method takes more CPU time to compute a steady solution than a first-order one on the same mesh. But, nobody is claiming that first-order methods are more efficient than second-order ones: first-order methods take more CPU time to achieve the same level of accuracy than second-order ones, and a much finer mesh is usually needed. When it comes to high-order methods, the same basis of comparison must be used.

We cannot evaluate method efficiency on the basis of the cost on the same mesh. We must do it on the basis of the cost to achieve the same error. For example, if an error of one drag count (0.0001 in units of the drag coefficient) is required in an aerodynamic computation, a high-order method may be more efficient than a low-order one because the high-order method can achieve this error threshold on a much coarser mesh. Therefore, the only fair way to compare efficiency is to look at the computational cost to achieve the same level of accuracy or given the same CPU, what error is produced. On this basis, high-order methods are not necessarily expensive.

### Belief 2. High-order methods are not needed for engineering accuracy

CFD has undergone tremendous development as a discipline for three decades and is used routinely to complement the wind tunnel in the design of aircraft [21]. The workhorse production codes use second-order finite volume method (FVM), finite difference method (FDM), or FEM. They are capable of running on small clusters with overnight turnaround time to achieve engineering accuracy (e.g., 5% error) for Reynolds-averaged Navier–Stokes (RANS) simulations. There was much excitement when the CFD community moved from first-order to second-order methods as the solution accuracy showed significant improvement. The reason is that when the mesh size and time step are reduced by half, the computational cost increases by a factor of roughly 16 (three spatial dimensions and one time dimension). Therefore, to reduce the error by a factor of 4, the DOFs increase by a factor of 256 for a first-order method and only 16 for a second-order one.

Whereas second-order methods have been the workhorse for CFD, there are still many flow problems that are too expensive or out of their reach. One such problem is the flow over a helicopter. The aerodynamic loading on the helicopter body is strongly influenced by the tip vortices generated by the rotor. These vortices travel many revolutions before hitting the body. It is critical that these vortices be resolved for a long distance to obtain even an engineering accuracy level prediction of the aerodynamic forces on the helicopter body. Because first-order and second-order methods strongly dissipate unsteady vortices, the mesh resolution requirement for the flow makes such a simulation too expensive even on modern supercomputers. The accurate resolution of unsteady vortices is quite a stringent requirement similar to that encountered in computational aeroacoustics (CAA) where broadband acoustic waves need to propagate for a long distance without significant numerical dissipation or dispersion errors. In the CAA community, high-order methods are used almost exclusively because of their superior accuracy and efficiency for problems requiring a high-level of accuracy [22]. Thus, for vortex-dominated flows, high-order methods are needed to accurately resolve unsteady vortices. Such flows play a critical role in the aerodynamic performance of flight vehicles.

Why should we stop at second-order accuracy? There is no evidence that second-order is the *sweet spot* in terms of the order of accuracy. The main reason that these methods are enjoying much success in engineering applications today is because of the research investment by the CFD community from the 1970s to the 1990s in making them efficient and robust. With additional research, high-order methods could become a workhorse for future CFD. Ultimately, the most efficient approach is to let the flow field dictate the local order of accuracy and grid resolution using *hp*-adaptation.

Another reason that second-order methods may not be accurate enough is the following. An acceptable solution error for one variable may lead to an unacceptable solution error for another. For example, a 5% error in velocity may translate into a 20% or higher error in skin friction depending on many factors such as Reynolds number, mesh density, and the method employed. As another example, for flows over helicopters, a 5% error in the drag coefficient may require that the strength of the tip vortices be resolved within 5% error over four to eight revolutions. In short, low-order methods cannot satisfy even engineering accuracy for numerous problems.

So much about CFD myths; now, let us turn to some justified concerns. The main reasons why high-order methods are not used in the design process include the following:

They are more complicated than low-order methods.

They are less robust and slower to converge to steady state because of the much reduced numerical dissipation.

They have a high memory requirement if implicit time stepping is employed.

Robust high-order mesh generators are not readily available.

In short, in spite of their potential, much remains to be done before high-order methods become a workhorse for CFD.

The main goals of the workshop on high-order methods are (i) to evaluate high-order and second-order methods in a fair manner for comparison and (ii) to identify remaining difficulties or pacing items. Concerning (i), we measure performance by comparing computational costs to achieve the same error. The workshop identified test cases and defined error and cost for a wide variety of methods and computers. In many cases, computational meshes were also provided.

The remainder of the paper is organized as follows. In Section 2, the motivation and history of the workshop are described. After that, the benchmark cases are presented together with how to compute errors and work units in Section 3. Section 5 depicts some representative results from the workshop to illustrate the current status in the development of high-order methods. Concluding remarks and pacing items for future work are described in Section 6.