# BVP Solver User Guide
This guide provides detailed instructions for using the Boundary Value Problem (BVP) solver in RustedSciThe with the `faer` linear algebra backend.
## Table of Contents
1. [Basic Setup](#basic-setup)
2. [Defining the ODE System](#defining-the-ode-system)
3. [Setting Boundary Conditions](#setting-boundary-conditions)
4. [Initial Guess](#initial-guess)
5. [Solver Parameters](#solver-parameters)
6. [Complete Examples](#complete-examples)
7. [Advanced Features](#advanced-features)
## Basic Setup
First, import the necessary types and functions:
```rust
use crate::numerical::BVP_sci::BVP_sci_faer::solve_bvp;
use faer::col::Col;
use faer::mat::Mat;
type faer_col = Col<f64>;
type faer_dense_mat = Mat<f64>;
```
## Defining the ODE System
The ODE system is defined as a function that takes three parameters:
- `x`: mesh points (faer_col)
- `y`: solution values at mesh points (faer_dense_mat, shape: n×m)
- `p`: parameters (faer_col)
Returns: derivatives (faer_dense_mat, shape: n×m)
### Example: Second-order ODE y'' = -y
```rust
```rust
// y'' = y^3 - sin(x)
let fun = |x: &faer_col, y: &faer_dense_mat, _p: &faer_col| {
let mut f = faer_dense_mat::zeros(2, y.ncols());
for j in 0..y.ncols() {
let y1 = *y.get(0, j);
*f.get_mut(0, j) = *y.get(1, j);
*f.get_mut(1, j) = y1 * y1 * y1 - x[j].sin();
}
f
};
```
#### Parametric System
```rust
// y'' + p[0]*y' + p[1]*y = 0
let fun = |_x: &faer_col, y: &faer_dense_mat, p: &faer_col| {
let mut f = faer_dense_mat::zeros(2, y.ncols());
for j in 0..y.ncols() {
*f.get_mut(0, j) = *y.get(1, j);
*f.get_mut(1, j) = -p[1] * *y.get(0, j) - p[0] * *y.get(1, j);
}
f
};
```
## Setting Boundary Conditions
Boundary conditions are defined as a function that takes:
- `ya`: solution values at left boundary (faer_col)
- `yb`: solution values at right boundary (faer_col)
- `p`: parameters (faer_col)
Returns: boundary condition residuals (faer_col, length: n + k)
### Common Boundary Condition Types
#### Dirichlet Conditions
```rust
// y(0) = 1, y(1) = 0
```rust
// y(0) = y(1), y'(0) = y'(1)
let bc = |ya: &faer_col, yb: &faer_col, _p: &faer_col| {
faer_col::from_fn(2, |i| match i {
0 => ya[0] - yb[0], // y(0) = y(1)
1 => ya[1] - yb[1], // y'(0) = y'(1)
_ => 0.0,
})
};
```
#### Parameter Estimation
```rust
// y(0) = 0, y(1) = 1, ∫y dx = target_integral
let bc = |ya: &faer_col, yb: &faer_col, p: &faer_col| {
faer_col::from_fn(3, |i| match i {
0 => ya[0], // y(0) = 0
1 => yb[0] - 1.0, // y(1) = 1
2 => p[0] - target_integral, // parameter constraint
_ => 0.0,
})
};
```
## Initial Guess
The initial guess is a matrix of shape (n, m) where:
- n = number of equations
- m = number of mesh points
### Creating Initial Guess
#### Constant Values
```rust
let mut y = faer_dense_mat::zeros(2, 5);
for j in 0..5 {
*y.get_mut(0, j) = 1.0; // y1 = 1 everywhere
*y.get_mut(1, j) = 0.0; // y2 = 0 everywhere
}
```
#### Linear Interpolation
```rust
for j in 0..5 {
*y.get_mut(0, j) = x[j]; // linear from 0 to 1
*y.get_mut(1, j) = 1.0; // constant derivative
}
```
#### Analytical Approximation
```rust
// For y'' = -y with y(0)=0, y(π)=0, use sin(x) as initial guess
for j in 0..5 {
*y.get_mut(0, j) = x[j].sin();
*y.get_mut(1, j) = x[j].cos();
}
```
## Solver Parameters
### Basic Parameters
```rust
let result = solve_bvp(
&fun, // ODE function
&bc, // Boundary conditions
x, // Initial mesh
y, // Initial guess
None, // ODE Jacobian (optional)
None, // BC Jacobian (optional)
None, // Parameters (optional)
None, // Parameter bounds (optional)
1e-6, // BVP tolerance
1000, // Max iterations
2, // Max mesh refinements
None, // BC tolerance (optional)
);
```
### Parameter Descriptions
- **BVP tolerance** (`1e-6`): Controls accuracy of collocation residuals
- **Max iterations** (`1000`): Maximum Newton iterations per mesh
- **Max refinements** (`2`): Maximum number of adaptive mesh refinements
- **BC tolerance** (`None`): Boundary condition tolerance (defaults to BVP tolerance)
### Tolerance Guidelines
| Simple linear | 1e-8 | 1e-10 |
| Nonlinear smooth | 1e-6 | 1e-8 |
| Stiff/oscillatory | 1e-4 | 1e-6 |
| Parameter estimation | 1e-5 | 1e-7 |
## Complete Examples
### Example 1: Simple Linear BVP
```rust
use crate::numerical::BVP_sci::BVP_sci_faer::solve_bvp;
use faer::col::Col;
use faer::mat::Mat;
type faer_col = Col<f64>;
type faer_dense_mat = Mat<f64>;
fn solve_linear_bvp() {
// y'' = 0, y(0) = 0, y(1) = 1
// Exact solution: y = x
let fun = |_x: &faer_col, y: &faer_dense_mat, _p: &faer_col| {
let mut f = faer_dense_mat::zeros(2, y.ncols());
for j in 0..y.ncols() {
*f.get_mut(0, j) = *y.get(1, j); // y' = y2
*f.get_mut(1, j) = 0.0; // y2' = 0
}
f
};
let bc = |ya: &faer_col, yb: &faer_col, _p: &faer_col| {
faer_col::from_fn(2, |i| match i {
0 => ya[0], // y(0) = 0
1 => yb[0] - 1.0, // y(1) = 1
_ => 0.0,
})
};
// Initial mesh
let x = faer_col::from_fn(3, |i| i as f64 * 0.5);
// Initial guess
let mut y = faer_dense_mat::zeros(2, 3);
for j in 0..3 {
*y.get_mut(0, j) = x[j]; // linear guess
*y.get_mut(1, j) = 1.0; // constant derivative
}
let result = solve_bvp(
&fun, &bc, x, y, None, None, None, None,
1e-8, 1000, 0, None
);
match result {
Ok(res) => {
if res.success {
println!("Solution found!");
for i in 0..res.x.nrows() {
println!("x = {:.3}, y = {:.6}", res.x[i], res.y.get(0, i));
}
}
}
Err(e) => println!("Error: {}", e),
}
}
```
### Example 2: Nonlinear BVP with Parameters
```rust
fn solve_parametric_bvp() {
// y'' + p*y = 0, y(0) = 0, y(π) = 0, max(y) = 1
let fun = |_x: &faer_col, y: &faer_dense_mat, p: &faer_col| {
let mut f = faer_dense_mat::zeros(2, y.ncols());
for j in 0..y.ncols() {
*f.get_mut(0, j) = *y.get(1, j);
*f.get_mut(1, j) = -p[0] * *y.get(0, j);
}
f
};
let bc = |ya: &faer_col, yb: &faer_col, p: &faer_col| {
faer_col::from_fn(3, |i| match i {
0 => ya[0], // y(0) = 0
1 => yb[0], // y(π) = 0
2 => p[0] - 1.0, // p = 1 (parameter constraint)
_ => 0.0,
})
};
let x = faer_col::from_fn(5, |i| i as f64 * std::f64::consts::PI / 4.0);
let mut y = faer_dense_mat::zeros(2, 5);
for j in 0..5 {
*y.get_mut(0, j) = x[j].sin();
*y.get_mut(1, j) = x[j].cos();
}
let p = faer_col::from_fn(1, |_| 1.0); // Initial parameter guess
let result = solve_bvp(
&fun, &bc, x, y, None, None, Some(p), None,
1e-6, 1000, 2, None
);
match result {
Ok(res) => {
if res.success {
println!("Parameter found: p = {:.6}", res.p.as_ref().unwrap()[0]);
}
}
Err(e) => println!("Error: {}", e),
}
}
```
## Advanced Features
### Custom Jacobians
For better performance and convergence, provide analytical Jacobians:
```rust
let fun_jac = |x: &faer_col, y: &faer_dense_mat, p: &faer_col| {
let m = x.nrows();
let mut df_dy = Vec::new();
let mut df_dp = Vec::new();
for j in 0..m {
// Jacobian w.r.t. y at each mesh point
let mut jac_y = faer::sparse::SparseColMat::try_new_from_triplets(
2, 2,
&[
faer::sparse::Triplet::new(0, 1, 1.0), // ∂f1/∂y2 = 1
faer::sparse::Triplet::new(1, 0, -p[0]), // ∂f2/∂y1 = -p
]
).unwrap();
df_dy.push(jac_y);
// Jacobian w.r.t. parameters
let mut jac_p = faer::sparse::SparseColMat::try_new_from_triplets(
2, 1,
&[faer::sparse::Triplet::new(1, 0, -*y.get(0, j))] // ∂f2/∂p = -y1
).unwrap();
df_dp.push(jac_p);
}
(df_dy, Some(df_dp))
};
3, 2,
&[faer::sparse::Triplet::new(0, 0, 1.0)] // ∂bc1/∂ya1 = 1
).unwrap();
let dbc_dyb = faer::sparse::SparseColMat::try_new_from_triplets(
3, 2,
&[faer::sparse::Triplet::new(1, 0, 1.0)] // ∂bc2/∂yb1 = 1
).unwrap();
let dbc_dp = faer::sparse::SparseColMat::try_new_from_triplets(
3, 1,
&[faer::sparse::Triplet::new(2, 0, 1.0)] // ∂bc3/∂p = 1
).unwrap();
(dbc_dya, dbc_dyb, Some(dbc_dp))
};
// Use with solver
let result = solve_bvp(
&fun, &bc, x, y,
Some(&fun_jac), // ODE Jacobian
Some(&bc_jac), // BC Jacobian
Some(p), None, 1e-6, 1000, 2, None
);
```
### Mesh Refinement Strategy
The solver automatically refines the mesh when:
- Collocation residuals are too large
- Solution changes rapidly between mesh points
- Maximum refinements not exceeded
Control refinement with the `max_refinements` parameter:
- `0`: No refinement (fixed mesh)
- `1-3`: Conservative refinement
- `4-10`: Aggressive refinement (may be slow)
### Error Handling
```rust
match solve_bvp(&fun, &bc, x, y, None, None, None, None, 1e-6, 1000, 2, None) {
Ok(result) => {
match result.status {
0 => println!("Success: Solution converged"),
1 => println!("Warning: Max refinements reached"),
2 => println!("Error: Singular Jacobian"),
3 => println!("Error: BC tolerance not met"),
_ => println!("Error: Max iterations reached"),
}
}
Err(e) => println!("Solver error: {}", e),
}
```
This guide covers the essential aspects of using the BVP solver. For more complex problems, consider starting with simpler initial guesses and gradually increasing accuracy requirements.