Role-Oriented Code Generation in an Engine for Solving Hyperbolic PDE Systems

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• Adaptive mesh refinement: Technique to dynamically adjust grid resolution in different regions of the computational domain to capture features efficiently. "allowing for dynamic adaptive mesh refinement."
• ADER-DG: A high-order Arbitrary DERivative Discontinuous Galerkin scheme combining ADER time integration with DG spatial discretization for hyperbolic PDEs. "the ADER-DG scheme"
• Array of Structures (AoS): Memory layout where all fields of a data record are stored contiguously, which can hinder SIMD across multiple records. "This Array of Structure (AoS) data layout"
• Auto-vectorization: Compiler optimization that automatically converts scalar operations into SIMD instructions when feasible. "compiler auto-vectorization may be enabled."
• AVX2: Advanced Vector Extensions 2, a SIMD instruction set extension on x86 CPUs enabling 256-bit vector operations. "for more efficient AVX2 operations."
• Cache misses: Performance events when requested data is not found in a CPU cache level, causing slower memory access. "significant loss of performances due to cache misses"
• Cauchy–Kowalewski procedure: Analytic technique to replace time derivatives using spatial derivatives for linear PDEs in ADER methods. "the Cauchy-Kowalewski procedure."
• CCZ4: A formulation of Einstein’s field equations used in numerical relativity, here as a complex PDE system benchmark. "the Einstein equations from relativistic astrophysics (CCZ4)"
• CFL number: Courant–Friedrichs–Lewy stability parameter that restricts the timestep relative to mesh spacing and wave speeds. "The next time step size depends on the CFL number."
• Discontinuous Galerkin (DG): Finite element method using element-local polynomial approximations with discontinuities at element interfaces. "ADER Discontinuous Galerkin (DG) method"
• Domain Specific Language (DSL): Specialized input language for configuring PDE problems and solver options. "a Domain Specific Language (DSL) defined via JSON Schema"
• Exascale: Computing systems capable of at least 10¹⁸ operations per second, posing unique optimization challenges. "upcoming exascale architectures."
• Finite Volume (FV) scheme: Discretization that conserves fluxes through control-volume faces; robust for shocks. "shock capturing FV scheme"
• Godunov-type: Family of finite volume methods using Riemann solvers at cell interfaces for flux evaluation. "Godunov-type FV schemes."
• Hyperbolic PDE: Class of partial differential equations governing wave-like phenomena with finite propagation speeds. "hyperbolic systems of partial differential equations (PDEs)."
• Jinja2: Templating engine used to generate C++ code (kernels and glue) from parameterized templates. "Jinja2 template engine library."
• JSON Schema: Formal specification for validating JSON documents; used to define the solver’s input DSL. "defined via JSON Schema"
• Kernel (HPC): A performance-critical, self-contained computational routine implementing a numerical substep. "These kernels are the critical parts of ExaHyPE"
• LIBXSMM: Library generating highly optimized small-matrix multiplication kernels tailored to CPU microarchitectures. "For these we employ LIBXSMM"
• Model–View–Controller (MVC): Architectural pattern separating data (Model), generation logic (Controller), and code templates (View). "Model-View-Controller (MVC) architectural pattern"
• MESA-PD: Particle dynamics code employing template-based code generation within the waLBerla framework. "the MESA-PD particle dynamics code"
• Navier–Stokes equations: Governing equations for fluid dynamics modeling conservation of mass, momentum, and energy (compressible form here). "the compressible Navier-Stokes equations"
• Non-conservative fluxes: Terms in PDE systems that cannot be written as the divergence of a flux vector and require special treatment. "non-conservative fluxes"
• Peano framework: AMR mesh infrastructure based on tree-structured Cartesian grids used for ExaHyPE. "using the Peano framework"
• Picard iterations: Fixed-point iteration method to solve nonlinear systems arising in element-local predictor steps. "calculated using Picard iterations"
• Relaxed discrete maximum principle: Numerical admissibility criterion ensuring boundedness of polynomial solutions within a tolerance. "a relaxed discrete maximum principle in the sense of polynomials"
• Riemann problem: Initial-value problem with a piecewise-constant discontinuity used to compute inter-cell numerical fluxes. "to evaluate the Riemann problem on element faces."
• Riemann solver: Numerical method for resolving fluxes at discontinuities between cells based on local wave propagation. "We introduce a classical Riemann solver"
• SIMD: Single Instruction, Multiple Data; parallel execution model applying one instruction to multiple data elements. "low-level SIMD vectorization support."
• Shock capturing: Numerical techniques to stably resolve shock waves without spurious oscillations. "high resolution shock capturing FV scheme"
• Space-time predictor: Element-local ADER step computing a space–time polynomial approximation before interface corrections. "the space-time predictor."
• Structure of Array (SoA): Memory layout storing each field across all records contiguously, improving SIMD across records. "Structure of Array (SoA) layout"
• Tensor operations: Multidimensional linear algebra operations (e.g., contractions) arising from DG discretizations. "element-local tensor operations."
• UFL: Unified Form Language, a DSL for variational forms often paired with automated code generation. "such as UFL"
• Viscous flux: Flux contributions dependent on gradients (e.g., stress and heat conduction) in Navier–Stokes systems. "viscous flux terms"
• waLBerla framework: HPC software framework supporting modular, performance-portable simulation components. "waLBerla framework"

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