arXiv subjects

Jehanzeb Chaudhry

Publications and source records attributed to Jehanzeb Chaudhry.

5 recordsLinked to original sources

arXiv2026

A Weighted Upwind Vector Kinetic Lattice Boltzmann Method For Hyperbolic Conservation Laws

Vector kinetic lattice Boltzmann (VKLB) methods have recently emerged as a promising framework for solving hyperbolic partial differential equation (PDE) systems. VKLB discretizes Boltzmann-type equations using a discrete set of lattice velocities, enforces discrete moment constraints, and carefully defines equilibrium distribution functions. In this work, we introduce novel upwinded equilibrium distribution functions constructed from conservation variables and numerical fluxes derived via continuous flux vector splitting based on the eigendecomposition of the flux Jacobian. This formulation enables the weighted-upwind VKLB equilibrium to be applied broadly to general hyperbolic systems. The method is verified on a set of challenging hyperbolic systems that includes the shallow water, Euler and ideal magnetohydrodynamics (MHD) equations. The proposed method demonstrates improved stability, reduced error norms, and sharper shock resolution across increasingly complex verification and benchmark problems.

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arXiv2022

Error estimation for the time to a threshold value in evolutionary partial differential equations

We develop an \textit{a posteriori} error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals {of the solution}. Taylor's theorem and an adjoint-based \textit{a posteriori} analysis is used to derive computable and accurate error estimates in the case of semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.

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arXiv2022

Adjoint-based Adaptive Multi-Level Monte Carlo for Differential Equations

We present a multi-level Monte Carlo (MLMC) algorithm with adaptively refined meshes and accurately computed stopping-criteria utilizing adjoint-based a posteriori error analysis for differential equations. This is in contrast to classical MLMC algorithms that use either a hierarchy of uniform meshes or adaptively refined meshes based on Richardson extrapolation, and employ a stopping criteria that relies on assumptions on the convergence rate of the MLMC levels. This work develops two adaptive refinement strategies for the MLMC algorithm. These strategies are based on a decomposition of an error estimate of the MLMC bias and utilize variational analysis, adjoint problems and computable residuals.

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arXiv2022

A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations

We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of "local" problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete Parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the Parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our Time Parallel Algorithm (TPA) and Space-Time Parallel Algorithm (STPA) respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error.

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arXiv2019

A posteriori error analysis for Schwarz overlapping domain decomposition methods

Domain decomposition methods are widely used for the numerical solution of partial differential equations on high performance computers. We develop an adjoint-based a posteriori error analysis for both multiplicative and additive overlapping Schwarz domain decomposition methods. The numerical error in a user-specified functional of the solution (quantity of interest) is decomposed into contributions that arise as a result of the finite iteration between the subdomains and from the spatial discretization. The spatial discretization contribution is further decomposed into contributions arising from each subdomain. This decomposition of the numerical error is used to construct a two stage solution strategy that efficiently reduces the error in the quantity of interest by adjusting the relative contributions to the error.

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