Numerical solution to PDEs through different schemes (implicit or explicit) results in some error which are referred as dissipative or dispersive error for the solution of Hyperbolic or Parabolic PDE. How these are verified and minimised for more accurate results
We solve governing partial differential equations numerically in order to get solutions to complex fluid mechanics problems. In classical CFD using finite differences we first convert the p.d.e. to its discrete version by replacing each partial derivative term by a difference term using forward, backward or central differencing in time and space. The off-node and off-time level variables (like ui+1 & un+1) can be expanded as Taylor Series. After algebraic manipulations we arrange on the LHS of equation the difference form of the p.d.e. and on the RHS the other terms due to Taylor series. These left out terms on RHS contain delta_t, delta_x, delta_y along with the partial derivative terms of varying powers. Normally the terms on RHS are called the ‘truncation error terms’ and are neglected and we solve for the LHS difference equation representing the original p.d.e. Obviously how accurate and what type of results we get depends on the magnitude and nature of the discarded truncation error terms on the RHS. For the sake of better (convenient) physical interpretation all the partial time-derivatives occurring on the RHS are transformed into equivalent spatial partial derivatives, so that the truncation terms now have delta_t, delta_x, delta_y, constants and only spatial partial derivatives of various orders. The complete formulation (RHS + LHS) is called the ‘MODIFIED EQUATION’. As stated above the accuracy and the nature of the solutions we get depend on the truncation terms on the RHS.
The ACCURACY of the solution depends on the order of the delta_t, delta_x, delta_y present in the first term on RHS having lowest order of partial derivative. For example if we have delta_t and (delta_x)^2 terms in the very first term on RHS, our solution is said to be first order accurate in time and second order accurate in space.
The NATURE of the solution means whether the solution is DISSIPATIVE or DISPERSIVE. If the lowest order partial derivative term on the RHS is ‘even’ i.e. 2,4,6 etc. the solution is DISSIPATIVE in nature. Whereas if the lowest order partial derivative term on the RHS is ‘odd’ i.e. 1,3,5 etc. the solution is DISPERSIVE in nature. In a DISSIPATIVE solution the sharp gradients like shocks get smeared off due to DISSIPATION and appear as a ‘ramp’ rather than a ‘step’. If the solution on the other hand is DISPERSIVE, the solution gets violently wavy in the region of sharp gradients, for example at the corners of the shock solution. (See John D Anderson’s CFD Book page 236).
One of the most insidious sources of diffusion (dissipation) errors in the numerical solution of PDEs is associated with "convective" terms, e.g. of the form u*du/dx, if the differential is approximated with a first-order upwind approximation: u(i)*[u(i)-u(i-1)]/w. In fact, most journals (e.g. the ASME journals) will now not even accept papers whose numerical results are not formally at least 2nd-order accurate. It is now standard to use something like 3rd-order upwind discretization for such terms (much less diffusive). Formal analysis of these types of errors can be carried out by deriving the so-called modified equations, the method of which is discussed extensively in Tannehill, Anderson, and Pletcher (1997) and other authoritative books on numerical solution of PDEs.
We solve governing partial differential equations numerically in order to get solutions to complex fluid mechanics problems. In classical CFD using finite differences we first convert the p.d.e. to its discrete version by replacing each partial derivative term by a difference term using forward, backward or central differencing in time and space. The off-node and off-time level variables (like ui+1 & un+1) can be expanded as Taylor Series. After algebraic manipulations we arrange on the LHS of equation the difference form of the p.d.e. and on the RHS the other terms due to Taylor series. These left out terms on RHS contain delta_t, delta_x, delta_y along with the partial derivative terms of varying powers. Normally the terms on RHS are called the ‘truncation error terms’ and are neglected and we solve for the LHS difference equation representing the original p.d.e. Obviously how accurate and what type of results we get depends on the magnitude and nature of the discarded truncation error terms on the RHS. For the sake of better (convenient) physical interpretation all the partial time-derivatives occurring on the RHS are transformed into equivalent spatial partial derivatives, so that the truncation terms now have delta_t, delta_x, delta_y, constants and only spatial partial derivatives of various orders. The complete formulation (RHS + LHS) is called the ‘MODIFIED EQUATION’. As stated above the accuracy and the nature of the solutions we get depend on the truncation terms on the RHS.
The ACCURACY of the solution depends on the order of the delta_t, delta_x, delta_y present in the first term on RHS having lowest order of partial derivative. For example if we have delta_t and (delta_x)^2 terms in the very first term on RHS, our solution is said to be first order accurate in time and second order accurate in space.
The NATURE of the solution means whether the solution is DISSIPATIVE or DISPERSIVE. If the lowest order partial derivative term on the RHS is ‘even’ i.e. 2,4,6 etc. the solution is DISSIPATIVE in nature. Whereas if the lowest order partial derivative term on the RHS is ‘odd’ i.e. 1,3,5 etc. the solution is DISPERSIVE in nature. In a DISSIPATIVE solution the sharp gradients like shocks get smeared off due to DISSIPATION and appear as a ‘ramp’ rather than a ‘step’. If the solution on the other hand is DISPERSIVE, the solution gets violently wavy in the region of sharp gradients, for example at the corners of the shock solution. (See John D Anderson’s CFD Book page 236).
In addition to Rajeev's answer: dissipative errors (i.e. numerical diffusion) will act similar to natural diffusion, it will smear out gradients; dispersive errors will affect the propagation of signals, which will (in the presence of dispersion) not travel at the correct speed but at a different one. As example, you can analyse the action of a numerical scheme on the simple linear advection equation du/dt-c*du/dx=0. The exact solution transports an arbitrary front with speed c with no change in shape. If you introduce a numerical grid and some discretisation in time and space, and you analyse the transport of a sinus signal of a certain wavelength, you will find that particularly signals of short wavelength will be transported at too small speed. The shortest wavelength which can be resolved is the one according to Shannon's theorem with the wavelength of the computational grid. This is dispersion.
hi you can found a good physical explanation of dissipative or dispersive error and also on the amplitude error for the solution of Hyperbolic system in the course notes by P. Colella and E.G. : UC Berkeley E266A.
http://home.rzg.mpg.de/~bds/numerics/cfd-lectures.html
http://www.eecs.berkeley.edu/~colella/E266AFall2012/completeNotes.pdf
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The equivalence theorem of Lax for the convergence of a numerical solution is :
Consistency + Stability Convergence
where:
the Consistency is defined by : Lim T.E. = 0 when delta_t ->0, delta_x ->0, ..
the Stability can be analysed by the VonNewman analysis in order to get a condition on the included steps. This can show the disperssive or dissipative errors. Usually the second term included in the analysis after a first stage highlights these phenomena.
T.E. is for truncating error.
Roundoff errors are another source of instability.
I got the wavy flow depth for dam break on rigid flat bed (1 -D). I have used FVM (Finite Volume Method). What should I do now. Please suggest me
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