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Zebra rocks, characterized by their striking reddish‐brown stripes, rods, and spots of hematite (Fe‐oxide), showcase complex self‐organized patterns formed under far‐from‐equilibrium conditions. Despite their ease of recognition, the underlying mechanisms of pattern‐forming processes remain elusive. We introduce a novel advection‐dominated phase‐field model that effectively replicates the Liesegang‐like patterns observed in Zebra rocks. This numerical model leverages the concept of phase separation, a well‐established principle governing Liesegang phenomena in a two‐dimensional setting. Our findings reveal that initial solute concentration and fluid flow velocity are critical determinants in pattern morphologies. We quantitatively explain the spacing and width of a specific Liesegang‐like pattern category. Furthermore, the model demonstrates that vanishingly low initial concentrations promote the formation of oblique patterns, with inclination angles influenced by rock heterogeneity. Additionally, we establish a quantitative relationship between band thickness and geological parameters for orthogonal bands. This enables the characterization of critical geological parameters based solely on static patterns observed in Zebra rocks, providing valuable insights into their formation environments. The diverse patterns in Zebra rocks share similarities with morphologies observed on early Earth and Mars, such as banded iron formations and hematite spherules. Our model, therefore, offers a plausible explanation for the formation mechanisms of these patterns and presents a powerful tool for deciphering the geochemical environments of their origin. Plain Language Summary: Zebra rocks, known for their unique red and brown stripes and spots, hold clues to how similar patterns formed on early Earth and Mars. We have developed a new model to explain how these intriguing patterns form. The model suggests that the flow of fluids and the initial amount and location of dissolved iron‐oxide (rust) in water plays a big role in shaping the final Zebra rock design. The speed of the flow and the initial amount of rust can create different stripe and spot patterns, just like the ones in Zebra rocks. By studying these rocks, we can potentially decipher fluid flow scenarios of ancient environments on both Earth and Mars. Key Points: Patterns enable identification of flow velocity and concentration when observedFive Liesegang‐like patterns are replicated by varying flow velocities and initial concentrationsOne class of inclined patterns is identified in a very narrow flow velocity and low concentration regime [ABSTRACT FROM AUTHOR]

We propose a new class of high‐order time‐marching schemes with dissipation control and unconditional stability for parabolic equations. High‐order time integrators can deliver the optimal performance of highly accurate and robust spatial discretizations such as isogeometric analysis. The generalized‐α$$ \alpha $$ method delivers unconditional stability and second‐order accuracy in time and controls the numerical dissipation in the discrete spectrum's high‐frequency region. We extend the generalized‐α$$ \alpha $$ methodology to obtain high‐order time marching methods with high accuracy and dissipation control in the discrete high‐frequency range. Furthermore, we maintain the original stability region of the second‐order generalized‐α$$ \alpha $$ method in the new higher‐order methods; we increase the accuracy of the generalized‐α$$ \alpha $$ method while keeping the unconditional stability and user‐control features on the high‐frequency numerical dissipation. The methodology solves k>1,k∈ℕ$$ k>1,k\in \mathbb{N} $$ matrix problems and updates the system unknowns, which correspond to higher‐order terms in Taylor expansions to obtain (3/2k)th$$ \left(3/2k\right)\mathrm{th} $$‐order method for even k$$ k $$ and (3/2k+1/2)th$$ \left(3/2k+1/2\right)\mathrm{th} $$‐order for odd k$$ k $$. A single parameter ρ∞$$ {\rho}^{\infty } $$ controls the high‐frequency dissipation, while the update procedure follows the formulation of the original second‐order method. Additionally, we show that our method is A‐stable, and for ρ∞=0$$ {\rho}^{\infty }=0 $$ we obtain an L‐stable method. Furthermore, we extend this strategy to analyze the accuracy order of a generic method. Lastly, we provide numerical examples that validate our analysis of the method and demonstrate its performance. First, we simulate heat propagation; then, we analyze nonlinear problems, such as the Swift–Hohenberg and Cahn–Hilliard phase‐field models. To conclude, we compare the method to Runge–Kutta techniques in simulating the Lorenz system. [ABSTRACT FROM AUTHOR]

The laboratory measurements conducted on Vaca Muerta formation samples demonstrate stress-dependent elastic behavior and compaction under representative in situ conditions. The experimental results reveal that the analyzed samples display elastoplastic deformation and shear-enhanced compaction as primary plasticity mechanisms. These experimental findings contradict the expected linear elastic response anticipated before brittle failure, as reported in several studies on the geomechanical characterization of the Vaca Muerta formation. Therefore, we present a comprehensive laboratory analysis of Vaca Muerta formation samples showing their nonlinear elastic behavior and irrecoverable shear-enhanced compaction. Additionally, we calibrate an elastoplastic constitutive model based on these experimental observations. The resulting model accurately reproduces the observed phenomena, playing a pivotal role in geoengineering applications within the energy industry. [ABSTRACT FROM AUTHOR]