The density and temperature properties of the intergalactic medium (IGM) reflect the heating and ionization history during cosmological structure formation, and are primarily probed by the Lyman-alpha forest of neutral hydrogen absorption features in the observed spectra of background sources (Gunn & Peterson 1965). We present the methodology and initial results from the Cholla IGM Photoheating Simulation (CHIPS) suite performed with the Graphics Process Unit-accelerated Cholla code to study the IGM at high, uniform spatial resolution maintained over large volumes. In this first paper, we examine the IGM structure in CHIPS cosmological simulations that include IGM uniform photoheating and photoionization models where hydrogen reionization completes early (Haardt & Madau 2012) or by redshift z~6 (Puchwein et al. 2019). Comparing with observations of the large- and small-scale Lyman-alpha transmitted flux power spectra P(k) at redshifts 2 <~ z <~ 5.5, the relative agreement of the models depends on scale, with the self-consistent Puchwein et al. (2019) IGM photoheating and photoionization model in good agreement with the flux P(k) at k >~ 0.01 s/km at redshifts 2 <~ z <~ 3.5. On larger scales the P(k) measurements increase in amplitude from z~4.6 to z~2.2 faster than the models, and lie in between the model predictions at 2.2 <~ z <~ 4.6 for k~= 0.002-0.01 s/km. We argue the models could improve by changing the HeII photoheating rate associated with active galactic nuclei to reduce the IGM temperature at z~3. At higher redshifts z>~4.5 the observed flux P(k) amplitude increases at a rate intermediate between the models, and we argue that for models where hydrogen reionization completes late (z~5.5 - 6) resolving this disagreement will require inhomogeneous or patchy reionization. (Abridged)
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