Conservation form

<h2 id="general-form">General form</h2>
<p>Equations in conservation form take the form

∂
              ξ
            
            
              ∂
              t
            
          
        
        +
        
          ∇
        
        ⋅
        
          f
        
        (
        ξ
        )
        =
        0
      
    
    {\displaystyle {\frac {\partial \xi }{\partial t}}+{\boldsymbol {\nabla }}\cdot \mathbf {f} (\xi )=0}

for any conserved quantity 
  
    
      
        ξ
      
    
    {\displaystyle \xi }
  
, with a suitable function 
  
    
      
        
          f
        
      
    
    {\displaystyle \mathbf {f} }
  
. An equation of this form can be transformed into an <a href="/facts/Integral_equation/LCu4Imfq">integral equation</a>

d
            
              d
              t
            
          
        
        
          ∫
          
            V
          
        
        ξ
         
        d
        V
        =
        −
        
          ∮
          
            ∂
            V
          
        
        
          f
        
        (
        ξ
        )
        ⋅
        
          ν
        
         
        d
        S
      
    
    {\displaystyle {\frac {d}{dt}}\int _{V}\xi ~dV=-\oint _{\partial V}\mathbf {f} (\xi )\cdot {\boldsymbol {\nu }}~dS}

using the <a href="/facts/Divergence_theorem/pUGKF1N2">divergence theorem</a>. The integral equation states that the change rate of the integral of the quantity 
  
    
      
        ξ
      
    
    {\displaystyle \xi }
  
 over an arbitrary control volume 
  
    
      
        V
      
    
    {\displaystyle V}
  
 is given by the <a href="/facts/Flux/Aw3P5Bvb">flux</a> 
  
    
      
        
          f
        
        (
        ξ
        )
      
    
    {\displaystyle \mathbf {f} (\xi )}
  
 through the boundary of the control volume, with 
  
    
      
        
          ν
        
      
    
    {\displaystyle {\boldsymbol {\nu }}}
  
 being the outer <a href="/facts/Surface_normal/jhtLIhcu">surface normal</a> through the boundary. 
  
    
      
        ξ
      
    
    {\displaystyle \xi }
  
 is neither produced nor consumed inside of 
  
    
      
        V
      
    
    {\displaystyle V}
  
 and is hence conserved. A typical choice for 
  
    
      
        
          f
        
      
    
    {\displaystyle \mathbf {f} }
  
 is 
  
    
      
        
          f
        
        (
        ξ
        )
        =
        ξ
        
          u
        
      
    
    {\displaystyle \mathbf {f} (\xi )=\xi \mathbf {u} }
  
, with velocity 
  
    
      
        
          u
        
      
    
    {\displaystyle \mathbf {u} }
  
, meaning that the quantity 
  
    
      
        ξ
      
    
    {\displaystyle \xi }
  
 flows with a given velocity field.
</p><p>The integral form of such equations is usually the physically more natural formulation, and the differential equation arises from differentiation. Since the integral equation can also have non-differentiable solutions, the equality of both formulations can break down in some cases, leading to <a href="/facts/Weak_solution/udTTFPIX">weak solutions</a> and severe numerical difficulties in simulations of such equations.
</p>
<h2 id="example">Example</h2>
<p>An example of a set of equations written in conservation form are the <a href="/facts/Euler_equations_(fluid_dynamics)/EYW5lerl">Euler equations</a> of fluid flow:

∂
              ρ
            
            
              ∂
              t
            
          
        
        +
        ∇
        ⋅
        (
        ρ
        
          u
        
        )
        =
        0
      
    
    {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0}

∂
              ρ
              
                u
              
            
            
              ∂
              t
            
          
        
        +
        ∇
        ⋅
        (
        ρ
        
          u
        
        ⊗
        
          u
        
        +
        p
        
          I
        
        )
        =
        0
      
    
    {\displaystyle {\frac {\partial \rho \mathbf {u} }{\partial t}}+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} )=0}

∂
              E
            
            
              ∂
              t
            
          
        
        +
        ∇
        ⋅
        (
        
          u
        
        (
        E
        +
        p
        V
        )
        )
        =
        0
      
    
    {\displaystyle {\frac {\partial E}{\partial t}}+\nabla \cdot (\mathbf {u} (E+pV))=0}

</p><p>Each of these represents the <a href="/facts/Conservation_of_mass/fRu05SS1">conservation of mass</a>, <a href="/facts/Conservation_of_momentum/TTkGv5eY">momentum</a> and <a href="/facts/Conservation_of_energy/12CdN4Ik">energy</a>, respectively.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Conservation_law/6g1Ay2FR">Conservation law</a></li>
<li><a href="/facts/Lagrangian_and_Eulerian_specification_of_the_flow_field/sYLvgboC">Lagrangian and Eulerian specification of the flow field</a></li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Toro, E.F. (1999). <i>Riemann Solvers and Numerical Methods for Fluid Dynamics</i>. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-65966-8.</li>
<li>Randall J. LeVeque: <i>Finite Volume Methods for Hyperbolic Problems.</i> Cambridge University Press, Cambridge 2002, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-00924-3 (<i>Cambridge Texts in Applied Mathematics</i>).</li></ul>

Conservation form open-in-new

Conservation form