Equations

KdVEquation1D(; gravity, D = 1.0, eta0 = 0.0)

KdV (Korteweg-de Vries) equation in one spatial dimension. The equation is given by

\[\begin{aligned} \eta_t+\sqrt{g D} \eta_x+3 / 2 \sqrt{g / D} \eta \eta_x+1 / 6 \sqrt{g D} D^2 \eta_{x x x} &= 0. \end{aligned}\]

The unknown quantity of the KdV equation is the total water height $\eta$. The gravitational acceleration gravity is denoted by $g$ and the constant bottom topography (bathymetry) $b = \eta_0 - D$, where $\eta_0$ is the constant still-water surface and $D$ the still-water depth. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The KdV equation is only implemented for $\eta_0 = 0$.

The equations only support a flat bathymetry.

The KdV equation is first introduced by Joseph Valentin Boussinesq (1877) and rediscovered by Diederik Korteweg and Gustav de Vries in 1895.

The semidiscretization implemented here is a modification of the one proposed by Biswas, Ketcheson, Ranocha, and Schütz (2025) for the non-dimensionalized KdV equation $u_t + u u_x + u_{x x x} = 0.$

The semidiscretization is given by:

\[\begin{aligned} \eta_t + \sqrt{g D} D_1\eta + 1 / 2 \sqrt{g / D} \eta D_1 \eta + 1 / 2 \sqrt{g / D} D_1 \eta^2 + 1 / 6 \sqrt{g D} D^2 D_3\eta &= 0. \end{aligned}\]

where $D_1$ is a first-derivative operator, $D_3$ a third-derivative operator, and $D$ the still-water depth.

It conserves

• the total water mass (integral of $\eta$) as a linear invariant

and if upwind operators ($D_3 = D_{1,+} D_1 D_{1,-}$) or wide-stencil operators ($D_3 = D_1^3$) are used for the third derivative, it also conserves

• the entropy/total energy (integral of $U = 1/2\eta^2$)

for periodic boundary conditions.

• Diederik Korteweg and Gustav de Vries (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves DOI: 10.1080/14786449508620739
• Abhijit Biswas, David I. Ketcheson, Hendrik Ranocha and Jochen Schütz (2025) Traveling-Wave Solutions and Structure-Preserving Numerical Methods for a Hyperbolic Approximation of the Korteweg-de Vries Equation DOI: 10.1007/s10915-025-02898-x

BBMEquation1D(; gravity, D = 1.0, eta0 = 0.0, split_form = true)

BBM (Benjamin–Bona–Mahony) equation in one spatial dimension. The equation is given by

\[\begin{aligned} \eta_t + \sqrt{gD}\eta_x + \frac{3}{2}\sqrt{\frac{g}{D}}\eta\eta_x - \frac{1}{6}D^2\eta_{xxt} &= 0. \end{aligned}\]

The unknown quantity of the BBM equation is the total water height $\eta$. The gravitational acceleration gravity is denoted by $g$ and the constant bottom topography (bathymetry) $b = \eta_0 - D$, where $\eta_0$ is the constant still-water surface and $D$ the still-water depth. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The BBM equation is only implemented for $\eta_0 = 0$.

The equations only support a flat bathymetry.

The BBM equation is first described in Benjamin, Bona, and Mahony (1972). The semidiscretization implemented here is developed in Ranocha, Mitsotakis, and Ketcheson (2020) for split_form = true and in Linders, Ranocha, and Birken (2023) for split_form = false. If split_form is true, a split form in the semidiscretization is used, which conserves

• the total water mass (integral of $h$) as a linear invariant
• a quadratic invariant (integral of $1/2\eta(\eta - 1/6D^2\eta_{xx})$ or for periodic boundary conditions equivalently $\hat U = 1/2(\eta^2 + 1/6D^2\eta_x^2)$), which is called here energy_total_modified (and entropy_modified) because it contains derivatives of the solution

for periodic boundary conditions. If split_form is false the semidiscretization conserves

• the total water mass (integral of $h$) as a linear invariant
• the Hamiltonian (integral of $H = 1/4\sqrt{g/D}\eta^3 + 1/2\sqrt{gD}\eta^2$) (see hamiltonian)

for periodic boundary conditions.

• Thomas B. Benjamin, Jerry L. Bona and John J. Mahony (1972) Model equations for long waves in nonlinear dispersive systems DOI: 10.1098/rsta.1972.0032
• Hendrik Ranocha, Dimitrios Mitsotakis and David I. Ketcheson (2020) A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations DOI: 10.4208/cicp.OA-2020-0119
• Viktor Linders, Hendrik Ranocha and Philipp Birken (2023) Resolving entropy growth from iterative methods DOI: 10.1007/s10543-023-00992-w

BBMBBMEquations1D(; bathymetry_type = bathymetry_variable,
                  gravity, eta0 = 0.0)

BBM-BBM (Benjamin–Bona–Mahony) system in one spatial dimension. The equations for flat bathymetry are given by

\[\begin{aligned} \eta_t + ((\eta + D)v)_x - \frac{1}{6}D^2\eta_{xxt} &= 0,\\ v_t + g\eta_x + \left(\frac{1}{2}v^2\right)_x - \frac{1}{6}D^2v_{xxt} &= 0. \end{aligned}\]

The unknown quantities of the BBM-BBM equations are the total water height $\eta$ and the velocity $v$. The gravitational acceleration gravity is denoted by $g$ and the constant bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The BBM-BBM equations are only implemented for $\eta_0 = 0$.

Two types of bathymetry_type are supported:

• bathymetry_flat: flat bathymetry (typically $b = 0$ everywhere)
• bathymetry_variable: general variable bathymetry

For the general case of variable vathymetry the BBM-BBM equations are

\[\begin{aligned} \eta_t + ((\eta + D)v)_x - \frac{1}{6}(D^2\eta_{xt})_x &= 0,\\ v_t + g\eta_x + \left(\frac{1}{2}v^2\right)_x - \frac{1}{6}(D^2v_t)_{xx} &= 0. \end{aligned}\]

One reference for the BBM-BBM system can be found in Bona et al. (1998). The semidiscretization implemented here was developed for flat bathymetry in Ranocha et al. (2020) and generalized for a variable bathymetry in Lampert and Ranocha (2025). It conserves

• the total water mass (integral of $h$) as a linear invariant
• the total velocity (integral of $v$) as a linear invariant for flat bathymetry
• the total entropy/energy (integral of $U = 1/2 hv^2 + 1/2 g\eta^2$)

for periodic boundary conditions (see Lampert, Ranocha). For reflecting boundary conditions, the semidiscretization conserves

• the total water (integral of $h$) as a linear invariant
• the total entropy/energy (integral of $U = 1/2 hv^2 + 1/2 g\eta^2$)

Additionally, it is well-balanced for the lake-at-rest stationary solution, see Lampert and Ranocha (2025).

• Jerry L. Bona, Min Chen (1998) A Boussinesq system for two-way propagation of nonlinear dispersive waves DOI: 10.1016/S0167-2789(97)00249-2
• Hendrik Ranocha, Dimitrios Mitsotakis, David I. Ketcheson (2020) A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations DOI: 10.4208/cicp.OA-2020-0119
• Joshua Lampert, Hendrik Ranocha (2025) Structure-preserving numerical methods for two nonlinear systems of dispersive wave equations DOI: 10.1007/s44207-025-00006-3

SvaerdKalischEquations1D(; bathymetry_type = bathymetry_variable, gravity,
                         eta0 = 0.0, alpha = 0.0, beta = 1/3, gamma = 0.0)

Dispersive system by Svärd and Kalisch (2023) in one spatial dimension. The equations for variable bathymetry are given in conservative variables by

\[\begin{aligned} h_t + (hv)_x &= (\hat\alpha(\hat\alpha(h + b)_x)_x)_x,\\ (hv)_t + (hv^2)_x + gh(h + b)_x &= (\hat\alpha v(\hat\alpha(h + b)_x)_x)_x + (\hat\beta v_x)_{xt} + \frac{1}{2}(\hat\gamma v_x)_{xx} + \frac{1}{2}(\hat\gamma v_{xx})_x, \end{aligned}\]

where $\hat\alpha^2 = \alpha\sqrt{gD}D^2$, $\hat\beta = \beta D^3$, $\hat\gamma = \gamma\sqrt{gD}D^3$. The coefficients $\alpha$, $\beta$ and $\gamma$ are provided in dimensionless form and $D = \eta_0 - b$ is the still-water depth and eta0 is the still-water surface (lake-at-rest). The equations can be rewritten in primitive variables as

\[\begin{aligned} \eta_t + ((\eta + D)v)_x &= (\hat\alpha(\hat\alpha\eta_x)_x)_x,\\ v_t(\eta + D) - v((\eta + D)v)_x + ((\eta + D)v^2)_x + g(\eta + D)\eta_x &= (\hat\alpha v(\hat\alpha\eta_x)_x)_x - v(\hat\alpha(\hat\alpha\eta_x)_x)_x + (\hat\beta v_x)_{xt} + \frac{1}{2}(\hat\gamma v_x)_{xx} + \frac{1}{2}(\hat\gamma v_{xx})_x. \end{aligned}\]

The unknown quantities of the Svärd-Kalisch equations are the total water height $\eta$ and the velocity $v$. The gravitational acceleration gravity is denoted by $g$ and the bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$.

Currently, the equations only support a general variable bathymetry, see bathymetry_variable.

SvärdKalischEquations1D is an alias for SvaerdKalischEquations1D.

The equations by Svärd and Kalisch are presented and analyzed in Svärd and Kalisch (2025). The semidiscretization implemented here conserves

• the total water mass (integral of $h$) as a linear invariant
• the total momentum (integral of $h v$) as a nonlinear invariant for flat bathymetry
• the total modified entropy/energy (integral of $\hat U$/$\hat e$), which is called here entropy_modified and energy_total_modified

for periodic boundary conditions (see Lampert, Ranocha). Additionally, it is well-balanced for the lake-at-rest stationary solution, see Lampert and Ranocha (2025).

• Magnus Svärd, Henrik Kalisch (2025) A novel energy-bounded Boussinesq model and a well-balanced and stable numerical discretization arXiv: 2302.09924, DOI: 10.1016/j.jcp.2024.113516
• Joshua Lampert, Hendrik Ranocha (2025) Structure-preserving numerical methods for two nonlinear systems of dispersive wave equations DOI: 10.1007/s44207-025-00006-3

SvärdKalischEquations1D

Same as SvaerdKalischEquations1D.

SerreGreenNaghdiEquations1D(; bathymetry_type = bathymetry_variable,
                            gravity, eta0 = 0.0)

Serre-Green-Naghdi system in one spatial dimension. The equations for flat bathymetry are given by

\[\begin{aligned} h_t + (h v)_x &= 0,\\ h v_t - \frac{1}{3} (h^3 v_{tx})_x + \frac{1}{2} g (h^2)_x + \frac{1}{2} h (v^2)_x + p_x &= 0,\\ p &= \frac{1}{3} h^3 v_{x}^2 - \frac{1}{3} h^3 v v_{xx}. \end{aligned}\]

The unknown quantities of the Serre-Green-Naghdi equations are the total water height $\eta = h + b$ and the velocity $v$. The gravitational acceleration gravity is denoted by $g$ and the bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The total water height is therefore given by $\eta = h + b$.

Three types of bathymetry_type are supported:

• bathymetry_flat: flat bathymetry (typically $b = 0$ everywhere)
• bathymetry_mild_slope: variable bathymetry with mild-slope approximation
• bathymetry_variable: general variable bathymetry

For the mild-slope approximation, the Serre-Green-Naghdi equations are

\[\begin{aligned} h_t + (h v)_x &= 0,\\ h v_t - \frac{1}{3} (h^3 v_{tx})_x + \frac{1}{2} (h^2 b_x v_t)_x - \frac{1}{2} h^2 b_x v_{tx} + \frac{3}{4} h b_x^2 v_t + \frac{1}{2} g (h^2)_x + g h b_x + \frac{1}{2} h (v^2)_x + p_x + \frac{3}{2} \frac{p}{h} b_x &= 0,\\ p &= \frac{1}{3} h^3 v_{x}^2 - \frac{1}{3} h^3 v v_{xx} + \frac{1}{2} h^2 v (b_x v)_x. \end{aligned}\]

For the general case of variable bathymetry without mild-slope approximation, the Serre-Green-Naghdi equations are

\[\begin{aligned} h_t + (h v)_x &= 0,\\ h v_t - \frac{1}{3} (h^3 v_{tx})_x + \frac{1}{2} (h^2 b_x v_t)_x - \frac{1}{2} h^2 b_x v_{tx} + h b_x^2 v_t + \frac{1}{2} g (h^2)_x + g h b_x + \frac{1}{2} h (v^2)_x + p_x + \frac{3}{2} \frac{p}{h} b_x + \psi b_x &= 0,\\ p &= \frac{1}{3} h^3 v_{x}^2 - \frac{1}{3} h^3 v v_{xx} + \frac{1}{2} h^2 v (b_x v)_x,\\ \psi &= \frac{1}{4} h v (b_x v)_x. \end{aligned}\]

References for the Serre-Green-Naghdi system can be found in

• Serre (1953) Contribution â l'étude des écoulements permanents et variables dans les canaux DOI: 10.1051/lhb/1953034
• Green and Naghdi (1976) A derivation of equations for wave propagation in water of variable depth DOI: 10.1017/S0022112076002425

The semidiscretization implemented here conserves

• the total water mass (integral of $h$) as a linear invariant
• the total momentum (integral of $h v$) as a nonlinear invariant if the bathymetry is constant
• the total modified entropy/energy (integral of $\hat U$/$\hat e$), which is called here entropy_modified and energy_total_modified

for periodic boundary conditions (see Ranocha and Ricchiuto (2025)). Additionally, it is well-balanced for the lake-at-rest stationary solution, see

• Hendrik Ranocha and Mario Ricchiuto (2025) Structure-Preserving Approximations of the Serre-Green-Naghdi Equations in Standard and Hyperbolic Form DOI: 10.1002/num.70016

HyperbolicSerreGreenNaghdiEquations1D(; bathymetry_type = bathymetry_mild_slope,
                                      gravity,
                                      eta0 = 0.0,
                                      lambda)

Hyperbolic approximation of the Serre-Green-Naghdi system in one spatial dimension. The equations for flat bathymetry are given by

\[\begin{aligned} h_t + (h v)_x &= 0,\\ h v_t + \frac{1}{2} g (h^2)_x + \frac{1}{2} h (v^2)_x + \biggl( \frac{\lambda}{3} H (1 - H / h) \biggr)_x &= 0,\\ h w_t + h v w_x &= \lambda (1 - H / h),\\ H_t + H_x u &= w. \end{aligned}\]

The unknown quantities of the hyperbolized Serre-Green-Naghdi equations are the total water height $\eta = h + b$ and the velocity $v$. The gravitational acceleration gravity is denoted by $g$ and the bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The total water height is therefore given by $\eta = h + b$.

There are two additional variables $w \approx -h v_x$ and $H \approx h$ compared to the SerreGreenNaghdiEquations1D. In the original papers of Gavrilyuk et al., the variable $H$ is called $\eta$. Here, we use $\eta$ for the total water height and $H$ for auxiliary variable introduced in the hyperbolic approximation.

The relaxation parameter lambda ($\lambda$) introduced to obtain this hyperbolic approximation of the SerreGreenNaghdiEquations1D influences the stiffness of the system. For $\lambda \to \infty$, the hyperbolic Serre-Green-Naghdi equations converge (at least formally) to the original SerreGreenNaghdiEquations1D. However, the wave speeds of the hyperbolic system increase with increasing $\lambda$, so that explicit time integration methods become more expensive.

Two types of bathymetry_type are supported:

• bathymetry_flat: flat bathymetry (typically $b = 0$ everywhere)
• bathymetry_mild_slope: variable bathymetry with mild-slope approximation

For the mild-slope approximation, the Serre-Green-Naghdi equations are

\[\begin{aligned} h_t + (h v)_x &= 0,\\ h v_t + \frac{1}{2} g (h^2)_x + \frac{1}{2} h (v^2)_x + \biggl( \frac{\lambda}{3} H (1 - H / h) \biggr)_x + \biggl( g h + \frac{\lambda}{2} (1 - H / h) \biggr) b_x &= 0,\\ h w_t + h v w_x &= \lambda (1 - H / h),\\ H_t + H_x u + \frac{3}{2} b_x v &= w. \end{aligned}\]

References for the hyperbolized Serre-Green-Naghdi system can be found in

• Favrie and Gavrilyuk. A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves DOI: 10.1088/1361-6544/aa712d
• Busto, Dumbser, Escalante, Favrie, and Gavrilyuk. On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems DOI: 10.1007/s10915-021-01429-8

The semidiscretization implemented here conserves

• the total water mass (integral of $h$) as a linear invariant
• the total modified entropy/energy (integral of $\hat U$/$\hat e$), which is called here entropy_modified and energy_total_modified

for periodic boundary conditions (see Ranocha and Ricchiuto (2025)). Additionally, it is well-balanced for the lake-at-rest stationary solution, see

• Hendrik Ranocha and Mario Ricchiuto (2025) Structure-Preserving Approximations of the Serre-Green-Naghdi Equations in Standard and Hyperbolic Form DOI: 10.1002/num.70016

SainteMarieEquations1D(; bathymetry_type = bathymetry_mild_slope,
                         gravity,
                         eta0 = 0.0)

Sainte-Marie system in one spatial dimension (with parameter $\gamma = 2$ compared to the original literature). The equations are given by

\[\begin{aligned} h_t + (h v)_x &= 0,\\ h v_t + \frac{1}{2} g (h^2)_x + \frac{1}{2} h (v^2)_x + (h p)_x &= -(g h + 2 p) b_x,\\ h w_t + h v w_x &= 2 p,\\ v_x + (w - v b_x) / (h / 2) &= 0. \end{aligned}\]

The unknown quantities of the Sainte-Marie equations are the total water height $\eta = h + b$ and the velocity $v$. The gravitational acceleration gravity is denoted by $g$ and the bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The total water height is therefore given by $\eta = h + b$. The last two equations of the system determine the auxiliary variable $w$ and the non-hydrostatic pressure $p$.

The Sainte-Marie system is similar to the Serre-Green-Naghdi system, but with a different non-hydrostatic pressure term. In particular, it also requires solving elliptic equations to determine the time derivative of the velocity.

Two types of bathymetry_type are supported:

• bathymetry_flat: flat bathymetry (typically $b = 0$ everywhere)
• bathymetry_mild_slope: variable bathymetry with mild-slope approximation

References for the Sainte-Marie system can be found in

• Sainte-Marie (2011) Vertically averaged models for the free surface non-hydrostatic Euler system: derivation and kinetic interpretation DOI: 10.1142/S0218202511005118
• Bristeau, Mangeney, Sainte-Marie and Seguin (2015) An energy-consistent depth-averaged Euler system: derivation and properties DOI: 10.3934/dcdsb.2015.20.961
• Aïssiouene, Bristeau, Godlewski, Mangeney, Parés Madroñal and Sainte-Marie (2020) A two-dimensional method for a family of dispersive shallow water models DOI: 10.5802/smai-jcm.66
• Escalante, Dumbser and Castro (2019) An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes DOI: 10.1016/j.jcp.2019.05.035

HyperbolicSainteMarieEquations1D(; bathymetry_type = bathymetry_mild_slope,
                                   gravity,
                                   eta0 = 0.0,
                                   h0,
                                   alpha = 3.0)

Hyperbolic approximation of the Sainte-Marie system SainteMarieEquations1D in one spatial dimension (with parameter $\gamma = 2$ compared to the original literature) derived by Escalante, Dumbser and Castro (2019). The equations are given by

\[\begin{aligned} h_t + (h v)_x &= 0,\\ h v_t + \frac{1}{2} g (h^2)_x + \frac{1}{2} h (v^2)_x + (h p)_x &= -(g h + 2 p) b_x,\\ h w_t + h v w_x &= 2 p,\\ h p_t + h v p_x + h c^2 \bigl(v_x + (w - v b_x) / (h / 2)\bigr) &= 0. \end{aligned}\]

The unknown quantities of the Sainte-Marie equations are the total water height $\eta = h + b$ and the velocity $v$. The gravitational acceleration gravity is denoted by $g$ and the bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The total water height is therefore given by $\eta = h + b$.

There are two additional variables compared to the SainteMarieEquations1D: the auxiliary variable $w \approx -h v_x / 2 + v b_x$ and the non-hydrostatic pressure $p$. In the formal limit $c^2 \to \infty$, the hyperbolic approximation recovers the original Sainte-Marie system. Escalante, Dumbser and Castro (2019) choose the hyperbolization parameter as $c = \alpha \sqrt{g h_0}$ for some background water height $h_0$. Thus, the hyperbolization parameter $c^2$ is set by the keyword arguments alpha ($\alpha$), gravity ($g$), and h0 ($h_0$). The larger the value of $\alpha$, the better the approximation of the original system, but also the stiffer the system. Typically, $\alpha$ should be chosen larger than 1 to get a good approximation of the original system, but the system also becomes stiffer for larger values of $\alpha$. Escalante, Dumbser and Castro (2019) often use the value $\alpha = 3$ in their numerical experiments, which is also the default value in HyperbolicSainteMarieEquations1D. For the special case $\alpha = 0$ and with initial conditions $p = 0 = w$, the system reduces to the classical shallow water equations.

Two types of bathymetry_type are supported:

• bathymetry_flat: flat bathymetry (typically $b = 0$ everywhere)
• bathymetry_mild_slope: variable bathymetry with mild-slope approximation

References for the Sainte-Marie system and its hyperbolization can be found in

• Sainte-Marie (2011) Vertically averaged models for the free surface non-hydrostatic Euler system: derivation and kinetic interpretation DOI: 10.1142/S0218202511005118
• Bristeau, Mangeney, Sainte-Marie and Seguin (2015) An energy-consistent depth-averaged Euler system: derivation and properties DOI: 10.3934/dcdsb.2015.20.961
• Aïssiouene, Bristeau, Godlewski, Mangeney, Parés Madroñal and Sainte-Marie (2020) A two-dimensional method for a family of dispersive shallow water models DOI: 10.5802/smai-jcm.66
• Escalante, Dumbser and Castro (2019) An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes DOI: 10.1016/j.jcp.2019.05.035

The energy-conserving semidiscretization is basically taken from the following reference, but adapted to the standard style of DispersiveShallowWater.jl to use the primitive variables q = (η, v, D, w, p). The semidiscretization implemented here is derived from their method with parameters $\alpha_1 = 2$, $\alpha_2 = 0$, $\alpha_3 = 0$, and $\alpha_4 = 0$.

• Artiano and Ranocha (2026) On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems DOI: 10.48550/arXiv.2603.18978