1. Riemann Solver

Implementation

The f-wave solver is split into separate private methods for each computation step: waveSpeeds, waveStrengths, flux, and the public netUpdates method.

The flux method computes the flux function \(f(q) = [hu, \; hu^2 + \frac{1}{2}g h^2]^T\) for a given state. The waveStrengths method uses the flux difference \(\Delta f = f(q_r) - f(q_l)\) and decomposes it into the eigenvectors by solving the 2x2 system to obtain the coefficients \(\alpha_1, \alpha_2\).

A key difference to the existing Roe solver is the wave computation: the Roe solver multiplies the wave speed into the wave (e.g. \(\lambda \cdot \alpha\)), whereas the f-wave solver computes \(Z_p = \alpha_p \cdot r_p\) directly, i.e. \(Z_p[0] = \alpha_p\) and \(Z_p[1] = \alpha_p \cdot \lambda_p\).

Unit Tests

Eigenvalue Verification

Tested via [FWaveSpeeds]: given \(h_l = 10, u_l = -3\) and \(h_r = 9, u_r = 3\), the Roe averages are computed as \(h^{Roe} = 9.5\) and \(u^{Roe} = -0.079002...\), yielding the eigenvalues:

\[\lambda_1 = -9.7311093998375095, \quad \lambda_2 = 9.5731051658991654\]

Additionally, [FWaveFlux] verifies the flux computation for \((h, hu) = (10, -30)\) and \((h, hu) = (9, 27)\), and [FWaveStrengths] verifies the wave strength decomposition \([\alpha_1, \alpha_2]^T = R^{-1} \cdot \Delta f\) for the same input values, yielding \(\alpha_1 = 33.559...\) and \(\alpha_2 = 23.441...\).

Steady-State Test

Tested as part of [FWaveUpdates]: given \(q_l = q_r = [10, 0]^T\), the flux jump is zero:

\[\Delta f = f(q_r) - f(q_l) = 0\]

Therefore all wave strengths and net-updates are zero up to machine precision:

\[A^- \Delta Q = A^+ \Delta Q = 0\]

Supersonic Case

Two supersonic cases are tested as part of [FWaveUpdates]:

Supersonic right (\(\lambda_1, \lambda_2 > 0\)): given \(h_l = 1, hu_l = 50\) and \(h_r = 2, hu_r = 100\), the Roe velocity \(u^{Roe} = 50\) exceeds the wave speed \(\sqrt{g \cdot h^{Roe}} = \sqrt{9.80665 \cdot 1.5} \approx 3.834\), yielding \(\lambda_1 \approx 46.165 > 0\) and \(\lambda_2 \approx 53.835 > 0\). Both waves travel to the right, so the left net-update is zero:

\[A^- \Delta Q = 0, \quad A^+ \Delta Q = \Delta f\]

Supersonic left (\(\lambda_1, \lambda_2 < 0\)): symmetric case with negated momentum \(hu_l = -50, hu_r = -100\), yielding \(\lambda_1 \approx -53.835 < 0\) and \(\lambda_2 \approx -46.165 < 0\). Both waves travel to the left, so the right net-update is zero:

\[A^+ \Delta Q = 0, \quad A^- \Delta Q = \Delta f\]

Results & Visualizations

All unit tests passed as expected.

Individual Contributions

• Yannik Köllmann: Project setup (1.2), including forking the repository, build configuration with SCons, Doxygen setup, initial Catch2 integration and .gitignore setup. Implementation of the flux and waveStrengths methods for the f-wave solver. Adding release of Sphinx and Doxygen to GitHub Pages using GitHub Actions.

• Jan Vogt: CI/CD pipeline setup using GitHub Actions, including automated style checking with clang-format (clang-format --dry-run --Werror). Added .clang-format configuration (LLVM style, indent width 2) and .clangd config for IDE support in CLion. Extended the Nix shell with clang-tools and bear for local style checking and IDE support.

• Mika Brückner: Implementation of the waveSpeeds and netUpdates methods for the f-wave solver (task 1.3.1). Build configuration to include the FWave solver and tests (scons). Implementation of unit tests for waveSpeeds, flux, waveStrengths, and netUpdates, including steady-state and supersonic cases (task 1.3.2).