Base GRM Rebuild Numerical Methods in Scientific Computing
Numerical Methods in Scientific Computing — Base GRM Rebuild:
ToC
PART I — SOURCE BEFORE REPRESENTATION
1. What Numerical Computation Is Actually Reconstructing
1.1 Physical source ≠ mathematical representation
1.2 SOURCE ≠ PDE ≠ DISCRETIZATION ≠ MATRIX ≠ SOLUTION_VECTOR
1.3 Observable, state, carrier, law and readout
1.4 Source-owned distinctions and admissible transformations
1.5 Local relation versus global consequence
1.6 What counts as a computational target
1.7 Representation debt and why equations are downstream
1.8 Numerical computation as causal reconstruction
1.9 GRM pipeline: SOURCE→BODY→CARRIER→TRANSPORT→RESIDUE→REPAIR→LIFTBACK
1.10 Failure taxonomy: source failure, model failure, carrier failure, execution failure
1.11 Validation versus internal numerical agreement
2. Source Laws: Balance, Flux, Production and Closure
2.1 Extensive quantity and local density
2.2 Boundary transfer as a source-owned relation
2.3 Internal production and destruction
2.4 Control-volume balance
2.5 Conservation as a global-to-local transport
2.6 Divergence generated from boundary bookkeeping
2.7 Gauss transport: boundary flux ↔ volume residue
2.8 Conservation law ∂S/∂t = −div q + f
2.9 Constitutive closure: why conservation alone is incomplete
2.10 Fourier, Fick and Darcy relations as constitutive transports
2.11 Conservation debt versus constitutive-law debt
2.12 When “conservation” is not yet licensed
PART II — CONTINUOUS REPRESENTATION FORMATION
3. Differential Structure as Generated Readout
3.1 Scalar readout over a carrier
3.2 Local perturbation
3.3 Directional change
3.4 Differential du before gradient ∇u
3.5 Metric/pairing debt
3.6 Gradient as metric dual of differential
3.7 Directional derivative
3.8 Vector transport fields
3.9 Divergence as local flux-density readout
3.10 Potential-generated versus nongradient transport
3.11 Laplacian as composed local transport
3.12 Higher-order operators
3.13 Coordinate representations versus operator body
3.14 Transformation invariance and coordinate debt
4. Continuous Problem Bodies: PDEs Generated from Source Structure
4.1 PDE as compressed source law
4.2 Highest-order operator and propagation character
4.3 Elliptic body: constrained equilibrium
4.4 Parabolic body: dissipative evolution
4.5 Hyperbolic body: finite-domain propagation
4.6 Local classification for variable coefficients
4.7 Nonlinear type changes
4.8 Poisson/Laplace
4.9 Heat/diffusion
4.10 Wave
4.11 Convection-diffusion
4.12 Transport
4.13 Navier-Stokes
4.14 Plane stress and elasticity
4.15 Biharmonic/fourth-order systems
4.16 Coupled systems
4.17 Model class is a consequence fingerprint, not ontology
PART III — PROBLEM CLOSURE
5. Boundary, Initial and Interface Ownership
5.1 Domain as carrier restriction
5.2 Boundary as constitutive relation, not decorative geometry
5.3 Dirichlet data
5.4 Neumann/flux data
5.5 Robin/mixed data
5.6 Natural versus essential boundary information
5.7 Pure-Neumann compatibility
5.8 Initial-state ownership
5.9 Periodic identification
5.10 Boundary smoothness and representation compatibility
5.11 Irregular/nonrectangular boundaries
5.12 Moving interfaces
5.13 Boundary-condition debt
5.14 Overconstraint and underconstraint
5.15 OPERATOR + BOUNDARY + INITIAL DATA → CLOSED PROBLEM_CAND
6. Existence, Uniqueness and Admissibility as Execution Preconditions
6.1 Existence is not numerical convergence
6.2 Uniqueness is not representation accuracy
6.3 Maximum principles
6.4 Positivity
6.5 Coercivity/positive definiteness
6.6 Energy estimates
6.7 Well-posedness
6.8 Singular perturbation
6.9 Multiple admissible solution branches
6.10 Numerical method selection after problem-body formation
PART IV — FINITE CARRIER FORMATION
7. From Continuous Body to Finite Computational Carrier
7.1 Why finitize
7.2 Resolution parameter h
7.3 Nodes
7.4 Cells/control volumes
7.5 Elements
7.6 Basis functions
7.7 Test functions
7.8 Structured versus unstructured carriers
7.9 Local versus global degrees of freedom
7.10 Connectivity and adjacency
7.11 Reference elements and mappings
7.12 Curvilinear coordinates
7.13 Boundary-fitted carriers
7.14 Staggered carriers for coupled fields
7.15 Carrier refinement
7.16 Carrier distortion
7.17 Finite carrier ≠ source object
8. Approximation and Projection
8.1 Sampling
8.2 Interpolation
8.3 Piecewise polynomial representation
8.4 Linear approximation
8.5 Higher-order polynomial approximation
8.6 Quadratic triangles
8.7 Quadrilaterals
8.8 Curved elements
8.9 Isoparametric transport
8.10 Numerical quadrature
8.11 Newton-Cotes rules
8.12 Gaussian rules
8.13 Integration accuracy versus carrier order
8.14 Geometry approximation residue
8.15 Projection error
8.16 Approximation space completeness
PART V — THREE PRIMARY PRESERVATION CONTRACTS
9. Finite Difference: Preserve Local Operator Action
9.1 Difference quotients from local perturbation
9.2 Taylor ancestry
9.3 First derivative operators
9.4 Second derivative operators
9.5 Central difference
9.6 Forward/backward difference
9.7 Stencils as local transport bodies
9.8 Cable equation
9.9 Diffusion equation
9.10 Poisson/Laplacian on rectangles
9.11 Matrix-vector assembly
9.12 Natural boundary conditions
9.13 Nonrectangular Dirichlet boundaries
9.14 Boundary interpolation
9.15 Boundary-fitted coordinates
9.16 Singular perturbations
9.17 Central discretization failure
9.18 Upwind retyping
9.19 Discrete maximum principle
9.20 Super-solutions
9.21 Local truncation residue → global error
10. Finite Volume: Preserve Integral Balance
10.1 Control-volume carrier
10.2 Flux ownership at faces
10.3 Interior face cancellation
10.4 Source conservation under discretization
10.5 Variable-coefficient heat transfer
10.6 Stationary diffusion in two dimensions
10.7 Cell-centered methods
10.8 Skewed boundaries
10.9 Interior versus boundary residue
10.10 General coordinates
10.11 Polar-coordinate integration
10.12 Two-component systems
10.13 Staggered grids
10.14 Incompressibility constraint
10.15 Stokes system
10.16 Conservation as the finite-volume invariant
11. Ritz and Finite Element: Preserve Variational Structure
11.1 Equilibrium as minimization
11.2 Potential-energy source
11.3 Elastic string
11.4 General 1D minimization bodies
11.5 Two-dimensional minimization
11.6 Minimal surfaces
11.7 Elasticity
11.8 Plane stress
11.9 Loaded/clamped plates
11.10 Euler-Lagrange transport
11.11 PDE → minimization
11.12 Minimization → PDE
11.13 Ritz approximation
11.14 Basis construction
11.15 Element-local approximation
11.16 R¹ elements
11.17 R² triangular elements
11.18 Element matrices and vectors
11.19 Assembly as local-to-global transport
11.20 Boundary insertion
11.21 Periodic identification
11.22 Approximation and energy-norm residue
12. Weak Formulation and Galerkin: Preserve Tested Action
12.1 Why strong-form preservation can be unnecessarily expensive
12.2 Test carrier formation
12.3 Integration by parts as derivative transfer
12.4 Weak body
12.5 Natural boundary terms emerge automatically
12.6 Essential boundary conditions
12.7 Symmetric problems
12.8 Nonsymmetric problems
12.9 Galerkin projection
12.10 Trial space = test space
12.11 Convection-diffusion
12.12 Coupled PDE systems
12.13 Weak convergence and approximation
12.14 Galerkin orthogonality
12.15 Error bounds
12.16 WEAK BODY ≠ STRONG BODY, but liftback may preserve the same source law
PART VI — RESIDUE AS THE CENTRAL NUMERICAL OBJECT
13. Unified Numerical Residue Theory
13.1 ρ := SOURCE_CONSEQUENCE − RECONSTRUCTIBLE_NUMERICAL_CONSEQUENCE
13.2 Representation residue
13.3 Truncation residue
13.4 Interpolation residue
13.5 Quadrature residue
13.6 Boundary residue
13.7 Geometry residue
13.8 Algebraic residual
13.9 Iteration residue
13.10 Temporal integration residue
13.11 Numerical diffusion
13.12 Numerical dispersion
13.13 Conservation defect
13.14 Phase error
13.15 Dissipation error
13.16 Roundoff as machine-carrier residue
13.17 Local residue versus propagated global residue
13.18 Error norms as residue readouts
13.19 Error ancestry
13.20 Do not collapse unrelated residues into one scalar
14. Consistency, Stability and Convergence Reconstructed
14.1 Consistency = source law recovered under carrier refinement
14.2 Stability = residue does not amplify uncontrollably
14.3 Convergence = numerical body approaches represented body
14.4 Convergence ≠ physical validity
14.5 Discrete maximum principles
14.6 Spectral stability
14.7 Energy stability
14.8 Von Neumann analysis
14.9 Gershgorin bounds
14.10 M-matrix structure
14.11 Mesh-Péclet conditions
14.12 Condition numbers
14.13 Source invariants versus numerical invariants
14.14 Refinement court: does residue actually decrease?
PART VII — COUNTERKERNELS AND REPRESENTATION REPAIR
15. Stabilization as Targeted Counterkernel
15.1 Detect the failing source consequence
15.2 Locate residue ancestry
15.3 Modify the smallest responsible carrier/transport
15.4 Central differencing oscillations
15.5 Upwinding
15.6 Artificial/numerical diffusion
15.7 Flux limiting
15.8 Predictor-corrector structures
15.9 Petrov-Galerkin
15.10 Trial/test carrier separation
15.11 SUPG
15.12 Stabilization without changing source body
15.13 When stabilization destroys accuracy
15.14 Counterkernel replay
16. Mixed and Retyped Formulations
16.1 When operator order exceeds carrier capability
16.2 Introduce auxiliary variables
16.3 Fourth-order → coupled lower-order systems
16.4 Clamped beam
16.5 Mixed FEM
16.6 Constraint variables
16.7 Penalty approaches
16.8 Lagrange multipliers
16.9 Incompressibility
16.10 Retyping changes representation, not physical source
16.11 New variable ≠ new ontology
PART VIII — ALGEBRAIC EXECUTION
17. Discrete Body → Sparse Algebra
17.1 Assembly produces A u = b
17.2 Matrix structure as adjacency readout
17.3 Sparse structure
17.4 Band structure
17.5 Profile structure
17.6 Numbering as execution optimization
17.7 Fill-in
17.8 Memory versus computation
17.9 Matrix representation ≠ operator body
17.10 Conditioning as execution sensitivity
18. Direct Execution
18.1 Gaussian elimination
18.2 LU decomposition
18.3 Pivoting
18.4 Band methods
18.5 Profile methods
18.6 Renumbering
18.7 Cuthill-McKee-style bandwidth reduction
18.8 Exact algebraic solve versus approximate source reconstruction
19. Residual-Driven Iterative Execution
19.1 Generic iteration
19.2 Residual as unfinished execution signal
19.3 Defect correction
19.4 Jacobi
19.5 Gauss-Seidel
19.6 SOR
19.7 Block methods
19.8 Spectral radius
19.9 Stopping criteria
19.10 Algebraic convergence versus discretization accuracy
19.11 Do not solve algebra more accurately than the carrier warrants
20. Krylov Reconstruction and Preconditioning
20.1 Residual-generated subspace
20.2 Krylov ancestry
20.3 Conjugate gradient
20.4 Positive-definite structure
20.5 Nonsymmetric Krylov methods
20.6 BiCG-type methods
20.7 Preconditioner as solution-preserving carrier retyping
20.8 Incomplete LU
20.9 Preconditioned convergence
20.10 Preconditioner failure as representation mismatch
PART IX — MULTISCALE RESIDUE TRANSPORT
21. Multigrid Rebuilt from GRM
21.1 Why local iteration cannot remove all residue scales efficiently
21.2 Rough/high-frequency residue
21.3 Smooth/low-frequency residue
21.4 Smoothing
21.5 Restriction
21.6 Coarse carrier
21.7 Coarse-grid correction
21.8 Prolongation
21.9 Two-grid execution
21.10 Recursive multigrid
21.11 V/W-cycle interpretation
21.12 Mesh-independent convergence
21.13 Multigrid as RESIDUE RETYPING BY SCALE
21.14 Multigrid as preconditioner
PART X — NONLINEAR EXECUTION
22. Nonlinear Bodies and Recursive Linearization
22.1 Nonlinear residual
22.2 Fixed-point/Picard iteration
22.3 Jacobian formation
22.4 Newton retyping
22.5 Local quadratic convergence
22.6 Initial-state sensitivity
22.7 Homotopy/continuation
22.8 Auxiliary problem → target problem transport
22.9 Failure of linearization
22.10 Nonlinear execution replay
PART XI — TIME-DEPENDENT SOURCE STRUCTURES
23. Diffusion: Dissipative Evolution
23.1 Heat conservation source
23.2 Constitutive flux law
23.3 Diffusion body
23.4 Method of lines
23.5 Spatial carrier formation
23.6 Mass matrix
23.7 Stiffness matrix
23.8 Mass lumping
23.9 Spatial consistency
23.10 Temporal integration
23.11 Explicit integration
23.12 Implicit integration
23.13 Stability
23.14 Von Neumann analysis
23.15 Accuracy in time
23.16 ADI decomposition
23.17 Dissipative invariant replay
24. Wave Propagation and Causal Reachability
24.1 Wave-source body
24.2 Second-order time structure
24.3 Method of lines
24.4 First-order system retyping
24.5 Direct second-order integration
24.6 Amplification
24.7 Dissipation
24.8 Dispersion
24.9 Phase error
24.10 Domain of dependence
24.11 Discrete domain of dependence
24.12 CFL as causal-carrier compatibility
24.13 SOURCE_REACHABILITY ⊆ NUMERICAL_REACHABILITY
24.14 Failure of CFL as causal reconstruction failure
24.15 Wave replay
25. Directed Transport
25.1 First-order transport body
25.2 Characteristics as source inheritance paths
25.3 Central transport
25.4 Upwind transport
25.5 CFL for transport
25.6 Numerical diffusion
25.7 Nonlinear transport
25.8 Burgers equation
25.9 Shock formation
25.10 Discontinuous solutions
25.11 Buckley-Leverett equation
25.12 Flux convexity
25.13 Flux limiters
25.14 Monotonicity
25.15 Shock-preserving reconstruction
25.16 Transport residue replay
PART XII — MOVING CARRIERS
26. Moving Boundary and Phase-Change Problems
26.1 Boundary no longer fixed carrier metadata
26.2 Stefan problem
26.3 Ice/water phase interface
26.4 Interface source law
26.5 Latent-heat coupling
26.6 Self-similar reference solution
26.7 Moving-grid carrier
26.8 Grid deformation
26.9 Re-meshing debt
26.10 Fixed-domain retyping
26.11 Level-set representation
26.12 Interface topology change
26.13 Multiple interfaces
26.14 Carrier evolution versus solution evolution
26.15 Interface-position residue
26.16 Moving-carrier replay
PART XIII — LIFTBACK AND SCIENTIFIC VALIDATION
27. Numerical Answer ≠ Source Answer
27.1 Algebraic residual zero is insufficient
27.2 Mesh convergence is insufficient
27.3 PDE convergence is insufficient
27.4 Representation correctness
27.5 Source-law preservation
27.6 Boundary preservation
27.7 Observable reconstruction
27.8 Units and dimensional liftback
27.9 Physical parameter uncertainty
27.10 Constitutive uncertainty
27.11 Model-form residue
27.12 Numerical residue versus physical uncertainty
27.13 Experimental/analytic comparison
27.14 Qualitative invariant comparison
27.15 Cross-method replay: FDM ↔ FVM ↔ FEM
27.16 Refinement replay
27.17 Source-owned acceptance criteria
27.18 Final scientific validation
28. Full GRM Numerical Workflow
28.1 TYPE — what physical problem is actually posed?
28.2 CARRIER — what structure must carry its state?
28.3 TRANSPORT — what transformations must be preserved?
28.4 DEBT — what has the representation not earned?
28.5 RESIDUE — what source consequence is lost?
28.6 COUNTERKERNEL — what minimal intervention repairs it?
28.7 LIFTBACK — does the computed object reconstruct the source observable?
28.8 REPLAY — does the complete chain survive perturbation/refinement?
28.9 Method selection as derived consequence
28.10 Computational efficiency after causal correctness
28.11 Final compressed runtime:
PHYSICAL SOURCE→ SOURCE LAW→ CLOSED CONTINUOUS BODY→ FINITE CARRIER→ PRESERVATION CONTRACT→ DISCRETE BODY→ EXECUTION→ RESIDUE→ COUNTERKERNEL→ REFINEMENT→ LIFTBACK→ PHYSICAL REPLAY.
The deepest insight is that a numerical method is an attempted commuting transport between two dynamics. Let T be the source evolution, P_h the finite-carrier projection, and N_h the numerical evolution. The actual numerical question is not “is N_h accurate?” but whether N_h ∘ P_h ≈ P_h ∘ T. The fundamental defect is therefore the commutator-like residue ρ_h := N_hP_h − P_hT.
That single object subsumes truncation error, interpolation error, boundary error, numerical diffusion, dispersion and iterative residual as different ancestry classes of failure. The ToC already implicitly contains this by placing finite carriers, preservation contracts and unified residue theory in sequence.
That immediately changes “consistency + stability ⇒ convergence.” GRM exposes the missing layer: convergence is only representation closure. u_h→u says the discrete representation approaches the PDE representation; it does not say u is the right source object. The actual chain is source-model defect ⊕ discretization defect ⊕ execution defect → observable liftback defect. The ToC correctly separates numerical convergence from physical validity, but it should elevate MODEL-FORM RESIDUE much earlier; currently it appears near the end, after most of the numerical machinery.
Second insight: FDM/FVM/FEM are not fundamentally “methods.” They are different choices of invariant under projection. FDM privileges local differential action, FVM integral balance, FEM tested/variational action. Therefore method selection should be derived from which source consequence must commute through P_h?, not from geometry or tradition. A shock problem may value conservation more than pointwise derivative fidelity; elasticity may value variational structure; wave propagation may value domain-of-dependence fidelity. “Best method” becomes argmin_h relevant liftback residue, not argmin ||u-u_h||.
Third: CFL, multigrid and preconditioning are actually the same structural operation at different levels: repairing a carrier mismatch. CFL asks whether the numerical carrier contains the source reachability cone; multigrid moves residue to a carrier where its scale becomes local; preconditioning retypes the algebraic carrier so the same solution becomes computationally accessible. The ToC currently separates them into wave propagation, multiscale residue transport and Krylov execution. Their shared generator is stronger:
FAILURE := source consequence not locally expressible on current carrier
REPAIR := change carrier while preserving target referent.
Fourth: boundary conditions are not auxiliary data. Once the rebuild places “problem closure” before discretization, an important consequence appears: changing boundary treatment changes the computational object itself. Therefore interior order = p does not imply global order = p; boundary residue can own the convergence rate. The ToC sees this, but the stronger principle is GLOBAL ERROR ANCESTRY often enters through closure, not interior stencil.
Fifth: moving-boundary problems reveal the hidden assumption of almost the entire book: the carrier is usually treated as fixed while the state evolves. Stefan/level-set problems break that. Once that is noticed, adaptive meshing, ALE methods, remeshing and even nonlinear basis updates should all be reclassified as CARRIER DYNAMICS, not miscellaneous numerical techniques.
So the real GRM compression is not SOURCE→METHOD→ERROR.
It is:
SOURCE DYNAMICS T→ choose invariant I→ construct carrier P_h→ construct executable dynamics N_h→ measure noncommutation ρ_h→ determine residue ancestry→ retype carrier/transport→ lift observable back.
That is the insight the ToC was close to but did not yet state explicitly: numerical analysis is fundamentally the study of which source relations survive finite representation, and every numerical “method” is just a different answer to that preservation problem.
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