Wavelet Methods in Computational Combustion

Discretisation schemes based on the use of wavelet methods offer many potential advantages for the numerical simulation of combustion. In many cases of interest, flame structures are thin relative to the largest length scales of the problem and most length scales of the flow field, and so lend themselves to simulation using adaptive-mesh methods. Wavelet methods are naturally adaptive, in that the coefficients of the wavelet transform are non-zero only in regions where there is significant variation present in the solution. Hence, simple thresholding can be employed to make valuable savings in storage and in execution time. In this chapter, the basic principles of wavelet methods are established. Orthogonal and biorthogonal wavelet formulations are described and their advantages and disadvantages are discussed. An illustration of a wavelet-based discretisation scheme is provided using the Navier-Stokes momentum equation as an example. The same wavelet approach is applied to the simulation of a one-dimensional laminar premixed flame for which an asymptotic solution exists. Comparisons are made between the computational and analytical results and the accuracy of the wavelet approach is assessed. Extensions to higher dimensions are discussed. Finally, the current state of development of wavelet methods is outlined and conclusions are drawn.