# Discretization

Source: https://help.sim-flow.com/documentation/panels/discretization
Last updated: 2026-10-14

## Introduction

![img 5959ff39a1474](https://help.sim-flow.com/documentation/panels/images/discretization/img_5959ff39a1474.png)

To calculate the flow field, it is required to solve the [Navier-Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations) inside the flow domain. This is a system of partial differential equations which cannot be solved analytically in general and needs to be converted into a system of linear equations. The process of converting partial differential equations into a linear equation system is called discretization and can use multiple different schemes.

In the **Discretization** panel, one selects discretization schemes that are most suitable for the given simulation type. Usually, when selecting a discretization scheme, there is a trade-off between accuracy and stability, which may require some testing to select the optimal scheme.

## Time Discretization

![img 5959ff8c1faaa](https://help.sim-flow.com/documentation/panels/images/discretization/img_5959ff8c1faaa.png)

The **Time** discretization tab defines how terms of the form \(\frac{\partial\Psi}{\partial t}\) need to be discretized.

**Time Discretization Schemes:**

- **Steady State** - terms with temporal derivatives are not discretized, which is usually used in steady-state simulation
- **Implicit Euler** - basic Euler scheme (first-order), uses only the current and previous time-step values
- **Crank-Nicolson** - [Crank-Nicolson](https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method) scheme (second-order), requires specification of an "off-centering" coefficient (1 - pure Crank-Nicolson, 0 - pure Implicit Euler)
- **Backward** - backward-differencing scheme (second-order), uses the current and two previous time-step values
- **Local Euler** - local time-step Euler scheme (first-order), local time-step field is looked up from the database. This scheme should only be used for steady-state computations using transient codes where local time-stepping is preferable to under-relaxation for transport consistency reasons.
- **Bounded** - option for selecting the bounded form of the underlying scheme. Boundedness is achieved by subtracting \(\frac{\partial \varphi}{\partial t} \cdot \Psi\) which is non-conservative if \(\frac{\partial \rho}{ \partial t} \neq 0\) but conservative otherwise.

## Convection (Advection) Discretization

![img 5959ffc804b1e](https://help.sim-flow.com/documentation/panels/images/discretization/img_5959ffc804b1e.png)

The **Convection** discretization tab defines how terms of the form \(\nabla \cdot(\vec U\cdot \Psi)\) need to be discretized. Since usually different equations require different treatment, one needs to define a convection scheme for each individual equation.

### Defining Convection Scheme

![img 595a000511337](https://help.sim-flow.com/documentation/panels/images/discretization/img_595a000511337.png)

To specify the convection scheme for a given field (for example velocity), one needs to click the button next to the name of the equation. This will cause a panel with available options to expand. When the new scheme is selected, the label of the button will change and display the new scheme’s name.

**Convection Schemes:**

- **Upwind** - upwind differencing scheme (first-order, bounded)
- **Linear Upwind** - linear upwind differencing scheme (second-order), calculates upwind weighting factors and also applies a gradient-based explicit correction
- **Linear** - central-differencing scheme (second-order, unbounded)
- **Limited Linear** - TVD limited linear differencing scheme (second-order)
- **LUST** - linear-upwind stabilized transport scheme (second-order), blended linear/linear-upwind scheme. For this discretization, velocity gradients need to be specified. This scheme is used for large eddy simulation (LES).
- **Cubic** - modified version of the Linear scheme which applies an explicit correction (second-order)
- **van Leer** - discretization scheme with van Leer flux limiter (second-order)
- **MUSCL** - [MUSCL](https://en.wikipedia.org/wiki/MUSCL_scheme) differencing scheme (second-order)
- **QUICK** - [quadratic-upwind](https://en.wikipedia.org/wiki/QUICK_scheme) differencing scheme (second-order)
- **Interface Compression** - modeled relative velocity used to compress the interface
- **DES Hybrid** - hybrid convection scheme for DES calculations. The scheme blends between a low-dissipative linear convection scheme within the LES region and a numerically more robust upwind scheme in the RAS region. It requires you to specify respectively velocity and length scales.

**Convection Scheme Options:**

- **Vector Specific** - this option enables a special version of the scheme dedicated to discretization of vector fields
- **Bounded** - option for selecting the bounded form of the underlying scheme. Boundedness is achieved by subtracting \((\nabla \cdot \rho U)\Psi\) which is non-conservative if \(\frac{\partial \rho}{ \partial t} \neq 0\) but conservative otherwise. Can be used for convection of bounded scalar properties in steady-state solvers to improve stability if insufficient convergence of the pressure equation causes temporary divergence of the flux field.
- **Skew Corrected** - skewness-corrected interpolation scheme that applies an explicit correction to an underlying scheme

## Gradient Discretization

![img 595a004b42393](https://help.sim-flow.com/documentation/panels/images/discretization/img_595a004b42393.png)

The **Gradients** tab defines discretization schemes used for calculating gradients.

### Defining Gradient Scheme

![img 595a008f3ee31](https://help.sim-flow.com/documentation/panels/images/discretization/img_595a008f3ee31.png)

The **Gradient** discretization scheme, once selected, is applied to all fields.

**Gradient Schemes:**

- **Gauss** - basic gradient scheme using face-interpolation and [Gauss' theorem](https://en.wikipedia.org/wiki/Divergence_theorem) (second-order)
- **Least Squares** - gradient scheme using least-squares (second-order)
- **Fourth** - gradient scheme using least-squares (fourth-order)

**Gradient Limiting Schemes:**

- **none** - no limiting is applied
- **Cell Limited** - scalar limiter based on limiting the extrapolated face values between the maximum and minimum cell and cell neighbor values and is applied to all components of the gradient
- **Face Limited** - scalar limiter based on limiting the extrapolated face values between the face-neighbor cell values and is applied to all components of the gradient

### Defining Surface Normal Gradient Scheme

![img 595a00c3006c8](https://help.sim-flow.com/documentation/panels/images/discretization/img_595a00c3006c8.png)

The **Surface Normal Gradient**, once selected, applies to all fields also when calculating the Laplace operator.

**Surface Normal Gradient Schemes:**

- **Corrected** - simple central-difference scheme with non-orthogonal correction
- **Uncorrected** - simple central-difference scheme without non-orthogonal correction
- **Orthogonal** - simple central-difference scheme without non-orthogonal correction
- **Face Corrected** - simple central-difference scheme with non-orthogonal correction
- **Limited** - scheme with limited non-orthogonal correction. The limiter is controlled by a coefficient with a value between 0 and 1 which when zero switches the limiter off and the scheme behaves as Corrected, and when set to 1 the limiter is calculated such that the non-orthogonal contribution does not exceed the orthogonal part.
- **Linear Fit** - centered fit scheme with linear polynomial and explicit correction
- **Quadratic Fit** - centered fit scheme with quadratic polynomial and explicit correction

## Interpolation

![img 595a00f40e4fe](https://help.sim-flow.com/documentation/panels/images/discretization/img_595a00f40e4fe.png)

The **Interpolation** tab defines the face interpolation scheme, which is going to be used. The selected scheme will also be used when calculating Laplace and divergence operators.

**Interpolation Schemes:**

- **Linear** - central-differencing interpolation
- **Cubic** - modified version of the Linear scheme which applies an explicit correction
- **Mid Point** - mid-point interpolation (weighting factors = 0.5) scheme