# Stiff ODE System with SUNDIALS CVODE + Ginkgo

Solves the Robertson chemical kinetics problem — a classic stiff ODE test —
using SUNDIALS CVODE (BDF method) with Ginkgo providing the linear solver
for the implicit Newton steps.

$$\frac{du_1}{dt} = -0.04 u_1 + 10^4 u_2 u_3$$
$$\frac{du_2}{dt} = 0.04 u_1 - 10^4 u_2 u_3 - 3 \times 10^7 u_2^2$$
$$\frac{du_3}{dt} = 3 \times 10^7 u_2^2$$

Initial conditions: $u(0) = (1, 0, 0)$. Integrated to $t = 4 \times 10^7$.

## Integration Overview

Same native SUNDIALS-Ginkgo interface as the [Poisson example](../poisson/doc):
- `sundials::ginkgo::Matrix<GkoCsr>` wraps the Jacobian
- `sundials::ginkgo::LinearSolver<Factory>` wraps the Ginkgo GMRES solver

CVODE calls Ginkgo's GMRES at every implicit time step to solve the Newton
system $(I - \gamma J) \delta = -F$.

## Step-by-Step

### Right-Hand Side

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

### Jacobian

The 3x3 analytical Jacobian is stored as CSR (9 entries):

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

### Ginkgo Setup

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

### Time Integration

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

## Building

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

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

## Expected Output

```text
  t            u1          u2           u3
  0.000000e+00  1.000000e+00  0.000000e+00  0.000000e+00
  4.000000e-01  9.851721e-01  3.386395e-05  1.479402e-02
  4.000000e+00  9.055187e-01  2.240476e-05  9.445890e-02
  4.000000e+01  7.158271e-01  9.185536e-06  2.841637e-01
  4.000000e+02  4.505186e-01  3.222901e-06  5.494781e-01
  4.000000e+03  1.832022e-01  8.942367e-07  8.167969e-01
  4.000000e+04  3.898339e-02  1.621769e-07  9.610164e-01
  4.000000e+05  4.938276e-03  1.984995e-08  9.950617e-01
  4.000000e+06  5.168100e-04  2.068296e-09  9.994832e-01
  4.000000e+07  5.203083e-05  2.081340e-10  9.999480e-01
```