# Steady-State Poisson with SUNDIALS KINSOL + Ginkgo

Solves the nonlinear equation $-u''(x) - e^{u(x)} = 0$ on $[0,1]$ with
$u(0) = u(1) = 0$ using SUNDIALS KINSOL (Newton iteration). Ginkgo provides
the linear solver for each Newton step via SUNDIALS' native Ginkgo interface.

## Integration Overview

SUNDIALS v6.4+ provides native Ginkgo wrappers:
- `sundials::ginkgo::Matrix<GkoCsr>` wraps a Ginkgo CSR matrix as `SUNMatrix`
- `sundials::ginkgo::LinearSolver<Factory>` wraps a Ginkgo solver factory as `SUNLinearSolver`

These plug directly into KINSOL's `KINSetLinearSolver()` — no custom glue code.

## Step-by-Step

### Problem Definition

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [problem-setup]
:end-before: [/problem-setup]
```

### Nonlinear Residual

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [residual-function]
:end-before: [/residual-function]
```

### Jacobian Assembly

The Jacobian callback fills the Ginkgo CSR matrix that lives inside the
`SUNMatrix` wrapper:

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [jacobian-function]
:end-before: [/jacobian-function]
```

### Ginkgo Setup

Create the Ginkgo executor, CSR matrix, and solver factory, then wrap
them for SUNDIALS:

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [ginkgo-setup]
:end-before: [/ginkgo-setup]
```

### KINSOL Solve

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [kinsol-solve]
:end-before: [/kinsol-solve]
```

## Building

Requires SUNDIALS v6.4+ built with `-DENABLE_GINKGO=ON`.

```bash
cmake -S . -B build --preset default
cmake --build build
./build/poisson
```

## Expected Output

```text
KINSOL converged with flag = 0

Solution at selected points:
  u(0.00990099) = 0.00539008
  u(0.108911) = 0.0537863
  u(0.207921) = 0.091845
  u(0.306931) = 0.119166
  u(0.405941) = 0.135454
  u(0.50495) = 0.140527
  u(0.60396) = 0.134328
  u(0.70297) = 0.116927
  u(0.80198) = 0.0885165
  u(0.90099) = 0.0494044
```