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Finite difference approximation, in addition to Taylor truncation errors,introduces numerical dispersion-and-dissipation errors into numerical solutionsof partial differential equations. We analyze a class of finite differenceschemes which are designed to minimize these errors (at the expense of formalorder of accuracy), and we analyze the interplay between the Taylor truncationerrors and the dispersion-and-dissipation errors during mesh refinement. Inparticular, we study the numerical dispersion relation of the fully discretizednon-dispersive transport equation in one and two space dimensions. We derivethe numerical phase error and the $L^2$-norm error of the solution in terms ofthe dispersion-and-dissipation error. Based on our analysis, we investigate theerror dynamics among various optimized compact schemes and the unoptimizedhigher-order generalized Pad\'e compact schemes. The dynamics shed light on theprinciples of designing suitable optimized compact schemes for a given problem.Using these principles as guidelines, we then propose an optimized scheme thatprescribes the numerical dispersion relation before finding the correspondingdiscretization. This approach produces smaller numericaldispersion-and-dissipation errors for linear and nonlinear problems, comparedwith the unoptimized higher-order compact schemes and other optimized schemesdeveloped in the literature. Finally, we discuss the difficulty of developingan optimized composite boundary scheme for problems with non-trivial boundaryconditions. We propose a composite scheme that introduces a buffer zone toconnect an optimized interior scheme and an unoptimized boundary scheme. Ournumerical experiments show that this strategy produces small $L^2$-norm errorwhen a wave packet passes through the non-periodic boundary.

Analysis and development of compact finite difference schemes with optimized numerical dispersion relations