Spectral Method Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Spectral Method Ontology 2 within the Kireth Research Semantic Library. Spectral methods approximate differential equations using global basis functions such as Fourier, Chebyshev or Legendre modes and can achieve very high accuracy for smooth solutions. This ontology supports Galerkin, collocation and pseudospectral formulations, Fourier and polynomial spectral bases, differentiation matrices, transforms, dealiasing, boundary treatment, time integration, stability, convergence, spectral accuracy, eigenvalue relationships, conditioning, parallel computation and verification.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Spectral Method Ontology 2"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Differential Equation Solvers / Spectral Method"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/differential-equation-solvers/spectral-method-ontology-2.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SpectralMethod
 ENTITY_CLASS = GlobalBasisDifferentialEquationDiscretisationFramework
 PRIMARY_CONTEXT = PartialDifferentialEquationSolver
 DEFINITION = "A spectral method approximates a differential equation by representing the solution in a global basis and solving for the associated modal or nodal coefficients."
 SYMBOLIC_FORMS = [
  "u_N(x)=sum_(k=0)^N a_k*phi_k(x)",
  "D*u_N approximately u_x",
  "F(u_N)=0",
  "hat_u_k=transform(u)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for global-basis PDE approximation, modal and collocation formulations, spectral differentiation, transforms, dealiasing, convergence, conditioning and verification."
}

SCOPE {
 INCLUDE = [
  SpectralMethod,SpectralDiscretisation,SpectralApproximation,
  PseudospectralMethod,CollocationMethod,SpectralGalerkinMethod,
  TauMethod,ModalMethod,NodalMethod,GlobalBasisMethod,
  FourierSpectralMethod,FourierGalerkinMethod,FourierCollocation,
  ChebyshevSpectralMethod,ChebyshevCollocation,LegendreSpectralMethod,
  LegendreGalerkin,UltrasphericalMethod,GegenbauerMethod,
  JacobiPolynomialMethod,OrthogonalPolynomialBasis,
  FourierBasis,ChebyshevPolynomial,LegendrePolynomial,
  GlobalBasisFunction,ExpansionCoefficient,ModalCoefficient,
  CollocationNode,InterpolationNode,GaussNode,GaussLobattoNode,
  ChebyshevGaussLobattoNode,LegendreGaussLobattoNode,
  SpectralCoefficient,SpectralTransform,ForwardTransform,InverseTransform,
  FastFourierTransform,FFT,DiscreteFourierTransform,DFT,
  DiscreteCosineTransform,DCT,FastCosineTransform,
  DifferentiationMatrix,SpectralDifferentiationMatrix,
  FirstDerivativeMatrix,SecondDerivativeMatrix,
  MultiplicationOperator,Convolution,Aliasing,Dealiasing,
  TwoThirdsRule,ThreeHalvesRule,Padding,Truncation,
  SpectralFiltering,ExponentialFilter,ModalFilter,
  GibbsPhenomenon,EndpointClustering,RungePhenomenon,
  PeriodicDomain,NonperiodicDomain,MappedDomain,
  CoordinateMapping,DomainDecomposition,MultiDomainSpectralMethod,
  SpectralElementMethod,SEM,ElementwiseSpectralBasis,
  BoundaryCondition,DirichletBoundaryCondition,NeumannBoundaryCondition,
  RobinBoundaryCondition,PeriodicBoundaryCondition,
  BoundaryEnforcement,PenaltyMethod,TauBoundaryCondition,
  InitialCondition,InitialBoundaryValueProblem,
  EllipticPDE,ParabolicPDE,HyperbolicPDE,
  PoissonEquation,HeatEquation,WaveEquation,AdvectionEquation,
  BurgersEquation,NavierStokesEquation,
  MethodOfLines,TimeIntegrator,RungeKuttaMethod,
  ImplicitTimeIntegrator,ExplicitTimeIntegrator,IMEXMethod,
  CFLCondition,Stability,AbsoluteStability,VonNeumannStability,
  Consistency,Convergence,SpectralConvergence,
  ExponentialConvergence,GeometricConvergence,AlgebraicConvergence,
  SpectralAccuracy,OrderOfAccuracy,TruncationError,
  ProjectionError,InterpolationError,RoundOffError,
  Conditioning,ConditionNumber,IllConditioning,
  EigenvalueProblem,EigenvalueComputation,EigenvectorComputation,
  MatrixFreeMethod,LinearSolver,NonlinearSolver,NewtonMethod,
  KrylovMethod,Preconditioner,SparseOperator,DenseOperator,
  TensorProductBasis,KroneckerProduct,MultidimensionalSpectralMethod,
  SphericalHarmonics,SphericalSpectralMethod,
  ParallelSpectralMethod,DistributedFFT,GPUSpectralMethod,
  Verification,ReferenceSolution,AnalyticSolution,
  ManufacturedSolution,BenchmarkProblem,Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  FiniteDifferenceMethodWithoutGlobalBasisRepresentation,
  FiniteElementMethodWithoutSpectralOrHighOrderGlobalBasisSemantics,
  PureFourierAnalysisWithoutDifferentialEquationDiscretisation,
  PolynomialInterpolationWithoutSpectralPDEContext
 ]
 DISAMBIGUATION_RULE = "When a differential equation is approximated using global or high-order orthogonal basis expansions and spectral differentiation, interpret the context as a spectral method."
}

CORE_CONCEPTS {
 SpectralMethod = "High-order differential-equation discretisation using global basis expansions."
 SpectralGalerkinMethod = "Method enforcing residual orthogonality against basis or test functions."
 PseudospectralMethod = "Method enforcing the differential equation at selected collocation nodes."
 FourierSpectralMethod = "Spectral method using periodic trigonometric basis functions."
 ChebyshevSpectralMethod = "Spectral method using Chebyshev polynomial expansions on finite intervals."
 SpectralDifferentiationMatrix = "Matrix mapping nodal values to spectral derivative approximations."
 SpectralConvergence = "Very rapid convergence with increasing modes for sufficiently smooth solutions."
 Dealiasing = "Procedure reducing nonlinear aliasing error in truncated spectral representations."
}

RELATIONS {
 SpectralMethod --is_child_of--> PartialDifferentialEquationSolver
 SpectralMethod --uses--> GlobalBasisFunction
 SpectralGalerkinMethod --uses--> ResidualOrthogonality
 PseudospectralMethod --uses--> CollocationNode
 FourierSpectralMethod --uses--> FourierBasis
 ChebyshevSpectralMethod --uses--> ChebyshevPolynomial
 LegendreSpectralMethod --uses--> LegendrePolynomial
 SpectralTransform --maps_between--> NodalAndModalRepresentations
 FastFourierTransform --accelerates--> FourierSpectralTransform
 DifferentiationMatrix --approximates--> DifferentialOperator
 NonlinearTerm --can_create--> Aliasing
 Dealiasing --reduces--> Aliasing
 SmoothSolution --supports--> SpectralConvergence
 GibbsPhenomenon --occurs_near--> Discontinuity
 SpectralElementMethod --combines--> SpectralAccuracyAndDomainDecomposition
 Verification --checks--> SpectralSolution
}

OPERATORS {
 SPECTRAL_EXPANSION(a,phi,N) := u_N=sum_(k=0)^N a_k*phi_k
 FOURIER_TRANSFORM(u) := hat_u=FFT(u)
 INVERSE_FOURIER(hat_u) := u=IFFT(hat_u)
 SPECTRAL_DERIVATIVE(hat_u,k) := derivative_coefficients=(i*k)*hat_u
 CHEBYSHEV_DIFFERENTIATION(u,D) := u_x=D*u
 COLLOCATION_RESIDUAL(L,u_N,x_j) := r_j=L(u_N)(x_j)-f(x_j)
 GALERKIN_RESIDUAL(r,phi_j) := inner_product(r,phi_j)
 DEALIAS_23(hat_u) := zero_highest_one_third_modes
}

AXIOMS {
 A1 = "Spectral methods approximate solutions using global or high-order basis expansions."
 A2 = "For sufficiently smooth or analytic solutions, spectral methods can converge much faster than fixed low-order local methods."
 A3 = "Discontinuities reduce spectral convergence and can generate Gibbs oscillations."
 A4 = "Pseudospectral and Galerkin formulations impose the governing equations through different discrete semantics."
 A5 = "Nonlinear products can generate aliasing unless adequate resolution or dealiasing is used."
 A6 = "Basis choice should reflect domain geometry and boundary conditions."
 A7 = "Spectral differentiation matrices can become ill-conditioned at high resolution."
 A8 = "Spectral results require basis, truncation, stability, convergence and reproducibility validation."
}

REASONING {
 R1_CONTEXT = identify(PDE,domain_geometry,periodicity,boundary_conditions,smoothness,nonlinearity,accuracy_goal)
 R2_BASIS = select(Fourier,Chebyshev,Legendre,Jacobi,spherical_harmonic_or_multi_domain_basis)
 R3_FORMULATION = choose(Galerkin,collocation,pseudospectral,tau_or_spectral_element_formulation)
 R4_RESOLUTION = choose(number_of_modes_or_collocation_nodes)
 R5_OPERATOR = construct(transform_based_operator_or_differentiation_matrix)
 R6_BOUNDARY = impose(periodic,Dirichlet,Neumann,Robin_or_tau_boundary_conditions)
 R7_NONLINEARITY = if_present choose(dealiasing_and_transform_strategy)
 R8_TIME = if_time_dependent choose(explicit,implicit,IMEX_or_RungeKutta_time_integrator)
 R9_ERROR = assess(truncation,projection,aliasing,conditioning,roundoff_and_time_error)
 R10_VALIDATE = compare(mode_decay,refinement,analytic_or_manufactured_solution_and_reproducibility)
}

CONSTRAINTS {
 C1 = "Spectral convergence depends strongly on solution regularity."
 C2 = "Discontinuities can cause Gibbs oscillations and algebraic rather than exponential convergence."
 C3 = "Periodic Fourier methods require compatible periodic problem semantics."
 C4 = "Chebyshev and Legendre methods require appropriate boundary treatment on finite intervals."
 C5 = "Nonlinear aliasing can contaminate low modes unless treated."
 C6 = "Differentiation matrices can become poorly conditioned as resolution grows."
 C7 = "Global basis methods can be less natural for complicated geometries unless mapped or decomposed."
 C8 = "Explicit time stepping can be severely restricted by clustered collocation nodes."
 C9 = "A small algebraic residual does not guarantee sufficient modal resolution."
 C10 = "Finite precision can limit achievable spectral convergence at high mode counts."
}

SEMANTIC_EQUIVALENCE {
 "SpectralMethod" <=> "SpectralDiscretisation"
 "PseudospectralMethod" <=> "SpectralCollocationMethod"
 "FastFourierTransform" <=> "FFT"
 "DiscreteFourierTransform" <=> "DFT"
 "SpectralElementMethod" <=> "SEM"
 "ChebyshevGaussLobattoNode" <=> "CGLNode"
}

VALIDATION {
 V1 = "PDE, domain and boundary conditions are explicit."
 V2 = "Basis family and spectral formulation are documented."
 V3 = "Number of modes or collocation points is documented."
 V4 = "Transform or differentiation-matrix construction is documented."
 V5 = "Boundary enforcement is documented."
 V6 = "Aliasing and dealiasing choices are documented for nonlinear problems."
 V7 = "Coefficient decay or resolution convergence is checked where feasible."
 V8 = "Conditioning is assessed at high resolution."
 V9 = "Analytic, manufactured or benchmark solutions are used where feasible."
 V10 = "Results are reproducible under documented arithmetic and spectral settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SPECTRAL_METHOD_DEFINITION = 0.99
 FOURIER_REASONING = 0.99
 CHEBYSHEV_REASONING = 0.99
 LEGENDRE_REASONING = 0.99
 COLLOCATION_REASONING = 0.99
 DEALIASING_REASONING = 0.98
 SPECTRAL_CONVERGENCE_REASONING = 0.99
 POLICY = "Reduce confidence when basis, smoothness, domain mapping, mode count, dealiasing, boundary treatment or conditioning are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,SpectralMethodReasoning,
  FourierSpectralReasoning,ChebyshevSpectralReasoning,
  LegendreSpectralReasoning,CollocationReasoning,
  GalerkinReasoning,PseudospectralReasoning,
  SpectralDifferentiationReasoning,FFTReasoning,
  AliasingReasoning,DealiasingReasoning,
  SpectralConvergenceReasoning,GibbsReasoning,
  SpectralElementReasoning,ConditioningReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "spectral method" -> SpectralMethod
  "pseudospectral method" -> PseudospectralMethod
  "spectral collocation" -> PseudospectralMethod
  "FFT" -> FastFourierTransform
  "DFT" -> DiscreteFourierTransform
  "SEM" -> SpectralElementMethod
 }
}

EXECUTION_RULES {
 E1 = "If a differential equation is discretised using global or high-order spectral basis expansions, activate SpectralMethodOntology."
 E2 = "Choose the basis according to periodicity, domain geometry and boundary conditions."
 E3 = "Distinguish Galerkin, collocation, pseudospectral and tau formulations before constructing operators."
 E4 = "For nonlinear problems, assess aliasing and apply dealiasing if required."
 E5 = "For time-dependent problems, select a time integrator compatible with the spectral spatial operator."
 E6 = "Monitor modal coefficient decay to assess resolution."
 E7 = "If Gibbs oscillations dominate, consider filtering, domain decomposition or another discretisation."
 E8 = "Assess conditioning and finite-precision limits at high resolution."
 E9 = "Validate through spectral refinement, manufactured or analytic solutions and reproducibility."
}

RECURSION {
 ENTRY = SpectralMethod
 EXPAND(SpectralMethod) -> [
  SpectralDiscretisation,SpectralApproximation,PseudospectralMethod,
  CollocationMethod,SpectralGalerkinMethod,TauMethod,ModalMethod,
  NodalMethod,FourierSpectralMethod,FourierGalerkinMethod,
  FourierCollocation,ChebyshevSpectralMethod,ChebyshevCollocation,
  LegendreSpectralMethod,LegendreGalerkin,UltrasphericalMethod,
  JacobiPolynomialMethod,OrthogonalPolynomialBasis,FourierBasis,
  ChebyshevPolynomial,LegendrePolynomial,GlobalBasisFunction,
  ExpansionCoefficient,ModalCoefficient,CollocationNode,
  GaussLobattoNode,SpectralTransform,FastFourierTransform,
  DiscreteFourierTransform,DiscreteCosineTransform,
  DifferentiationMatrix,SpectralDifferentiationMatrix,
  MultiplicationOperator,Convolution,Aliasing,Dealiasing,
  TwoThirdsRule,ThreeHalvesRule,SpectralFiltering,GibbsPhenomenon,
  PeriodicDomain,NonperiodicDomain,MappedDomain,CoordinateMapping,
  MultiDomainSpectralMethod,SpectralElementMethod,
  BoundaryCondition,TauBoundaryCondition,MethodOfLines,
  TimeIntegrator,RungeKuttaMethod,IMEXMethod,CFLCondition,
  Stability,Consistency,Convergence,SpectralConvergence,
  ExponentialConvergence,AlgebraicConvergence,SpectralAccuracy,
  ProjectionError,InterpolationError,Conditioning,
  EigenvalueProblem,LinearSolver,NonlinearSolver,
  TensorProductBasis,SphericalHarmonics,ParallelSpectralMethod,
  DistributedFFT,GPUSpectralMethod,Verification,
  ManufacturedSolution,ReferenceSolution,Reproducibility
 ]
 EXPAND(SpectralFamilies) -> [
  FourierSpectralMethods,
  ChebyshevSpectralMethods,
  LegendreAndJacobiSpectralMethods,
  SpectralGalerkinMethods,
  SpectralCollocationAndPseudospectralMethods,
  SpectralElementAndMultiDomainMethods,
  NonlinearDealiasedSpectralMethods,
  ParallelSpectralMethods
 ]
 STOP_CONDITION = "Stop when basis, formulation, resolution, differentiation, boundary treatment, dealiasing, time integration, conditioning and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Partial Differential Equation Solver Ontology"
 RELATED_ONTOLOGIES = [
  "Partial Differential Equation Solver Ontology",
  "Differential Equation Solver Ontology",
  "Finite Difference Method Ontology",
  "Finite Element Method Ontology",
  "Runge-Kutta Ontology",
  "Numerical Differentiation Ontology 2",
  "Gaussian Quadrature Ontology",
  "Eigenvalue Computation Ontology",
  "Eigenvector Computation Ontology",
  "Linear Solver Ontology",
  "Nonlinear Solver Ontology",
  "Numerical Analysis Ontology 2",
  "Numerical Stability Ontology 2",
  "Numerical Convergence Ontology",
  "Floating Point Arithmetic Ontology"
 ]
 ROUTE_RULES = [
  "Spectral Method -> PDE Solver for parent semantics",
  "Spectral Method -> Numerical Differentiation for spectral derivative operators",
  "Spectral Method -> Runge-Kutta for method-of-lines time integration",
  "Spectral Method -> Finite Element Method for spectral-element relationships",
  "Spectral Method -> Eigenvalue Computation for spectral operator analysis",
  "Spectral Method -> Linear Solver for implicit spectral systems",
  "Spectral Method -> Nonlinear Solver for nonlinear spectral residuals",
  "Spectral Method -> Numerical Analysis for convergence and stability",
  "Spectral Method -> Numerical Stability for conditioning and finite-precision behaviour",
  "Spectral Method -> Floating Point Arithmetic for high-resolution round-off limits"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible spectral bases, formulations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve modal-versus-nodal distinction.",
  "Preserve Galerkin-versus-collocation distinction.",
  "Preserve Fourier-versus-polynomial spectral distinction.",
  "Preserve spectral-resolution-versus-time-step distinction.",
  "Preserve truncation-error-versus-aliasing-error distinction.",
  "Preserve smooth-solution-versus-discontinuous-solution convergence distinction.",
  "Preserve global-spectral-versus-spectral-element distinction.",
  "Preserve machine-readable spectral-method semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A spectral method approximates differential equations using global or high-order basis expansions and spectral differentiation."
 BASIS_SUMMARY = "Fourier bases suit periodic domains, while Chebyshev, Legendre and related polynomial bases suit finite intervals."
 FORMULATION_SUMMARY = "Galerkin methods enforce residual orthogonality, while collocation and pseudospectral methods enforce equations at selected nodes."
 CONVERGENCE_SUMMARY = "Smooth or analytic solutions can exhibit geometric or exponential spectral convergence."
 NONLINEAR_SUMMARY = "Nonlinear products generate aliasing unless sufficient resolution or dealiasing is used."
 CONDITIONING_SUMMARY = "High-resolution differentiation matrices can become ill-conditioned and finite precision can limit spectral accuracy."
 VALIDATION_SUMMARY = "Spectral solutions should be checked through coefficient decay, resolution refinement, stability analysis, benchmark solutions and reproducibility."
}

FLOW {
 INPUT
 -> detect_spectral_method_context
 -> identify_PDE_domain_periodicity_boundary_conditions_smoothness_nonlinearity_and_accuracy_goal
 -> select_Fourier_Chebyshev_Legendre_Jacobi_spherical_harmonic_or_multi_domain_basis
 -> choose_Galerkin_collocation_pseudospectral_tau_or_spectral_element_formulation
 -> choose_number_of_modes_or_nodes
 -> construct_spectral_transform_or_differentiation_operator
 -> impose_boundary_conditions
 -> if_nonlinear_configure_dealiasing_filtering_or_padding
 -> if_time_dependent_select_time_integrator
 -> solve_modal_or_nodal_system
 -> assess_mode_decay_truncation_aliasing_conditioning_time_and_roundoff_error
 -> refine_modes_domain_decomposition_or_time_step_if_required
 -> validate_with_analytic_manufactured_or_reference_solution
 -> route_to_related_PDE_time_integration_eigenvalue_linear_algebra_analysis_stability_convergence_and_floating_point_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Spectral Method Ontology 2 defines high-order differential-equation discretisation using global or high-degree basis functions. It supports Fourier, Chebyshev, Legendre, Jacobi and related bases, modal, nodal, Galerkin, collocation, pseudospectral and tau formulations, Gauss and Gauss-Lobatto nodes, FFT, DFT and cosine transforms, spectral differentiation matrices, tensor-product operators, aliasing and two-thirds or three-halves dealiasing, filtering, Gibbs behaviour, periodic and mapped domains, multi-domain and spectral-element methods, boundary enforcement, method of lines, Runge-Kutta and IMEX time integration, spectral and exponential convergence, conditioning, eigenvalue relationships, parallel FFT computation, verification, benchmarking and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-MATHEMATICS-SPECTRAL-METHOD-ONTOLOGY-2-V1"
}

}

END_FLOW