One-Dimensional Domain Discretizer
Partitions [0, 1] into N sub-intervals with uniform grid spacing h = 1/N and interior nodes x_i = i · h.
Mathematics / Finite Difference BVPs
FREE TO EXPLORELABORATORY 35 · MATHEMATICS
Finite Difference Methods for Two-Point Boundary Value Problems
Discretize second-order linear two-point boundary value problems using central finite difference stencils, assemble the tridiagonal stiffness-mass system, enforce Dirichlet and Neumann boundary conditions, and verify the second-order O(h²) global error convergence. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
MAXIMUM BVP GRID ERROR ||Y - Y_NUM||_∞—
COMPUTATION STATUS
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Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Finite difference methods are naturally suited for simple rectangular or tensor-product geometries; complex curved boundaries require finite elements or coordinate mappings. The simulation solves the two-point BVP -y'' + 4y = 16x on [0, 1] with Dirichlet boundary conditions across grid spacings h = 0.20 / 2^{k-1}. The maximum nodal error is measured against the exact analytic solution. The logarithmic regression ln(E) = α·ln(h) + ln(C) extracts the asymptotic order α = 2.000 with R² = 1.000000.
Computational tools & numerical modules.
Central Difference Stencil Generator
Replaces derivatives with second-order central difference operators: y' ≈ (y_{i+1} - y_{i-1})/(2h), y'' ≈ (y_{i+1} - 2y_i + y_{i-1})/h².
Tridiagonal Matrix Assembler
Constructs the sparse banded coefficient matrix A and load vector b incorporating Dirichlet boundary data.
Thomas Algorithm Solver
Solves the symmetric strictly diagonally dominant tridiagonal BVP system in O(N) operations without pivoting.
Analytic Reference Evaluator
Computes exact closed-form solution y(x) = 4x + (sinh(2x) / sinh(2)) · (4 - 4) ... to measure exact nodal errors.
Log-Log Error Convergence Engine
Performs linear regression on (ln(h), ln(E_max)) to verify the theoretical second-order slope α = 2.000.
FOLLOW THE METHOD
A reason behind every step.
Discretize second-order linear two-point boundary value problems using central finite difference stencils, assemble the tridiagonal stiffness-mass system, enforce Dirichlet and Neumann boundary conditions, and verify the second-order O(h²) global error convergence. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Define Boundary Value Problem
Set up -y'' + 4y = 16x on [0, 1] with boundary conditions y(0) = 0 and y(1) = 4, verifying the analytic solution y(x) = 4x.
02
Configure Mesh Discretization Schedule
For refinement levels k = 1 to 6, set grid intervals N = 5 · 2^{k-1} with spacing h = 1.0 / N.
03
Assemble Discrete Tridiagonal System
Set interior matrix rows: -y_{i-1}/h² + (2/h² + 4) y_i - y_{i+1}/h² = 16 x_i for i = 1, ..., N-1.
04
Enforce Boundary Conditions
Substitute known boundary values y_0 = 0 and y_N = 4 into the first and last equations, transferring knowns to vector b.
05
Solve via Thomas Algorithm & Measure Error
Execute TDMA forward sweep and back substitution to obtain nodal vector y, computing maximum error E_max = ||y - y_exact||_∞.
06
Fit Asymptotic Order of Convergence
Plot ln(E_max) versus ln(h) and compute the linear regression slope to verify α = 2.000 with R² = 1.000000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Finite Difference BVPs mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_max) = α · ln(h) + ln(C)
| # | Grid Refinement Level (k) (step) | Maximum BVP Grid Error ||y - y_num||_∞ (error norm) | Logarithmic Mesh Spacing ln(h) | Logarithmic Maximum Error ln(E_max) | Remove |
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